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ELEMENTS 

OF  PROJECTIVE 

GEOMETKY 

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http://www.archive.org/details/elementsofprojecOOIingiala 


^ELEMENTS  OF 
PROJECTIVE  GEOMETRY 


BY 
GEORGE  HERBERT  LING 

GEORGE  WENTWORTH 

AND 

DAVID  EUGENE  SMITH 


GINN  AND  COMPANY 

BOSTON     •     NEW    YORK     •     CHICAGO     •     LONDON 
ATLANTA     •     DALLAS     •     COLIIMHIIS     •     SAN    KKANCISCO 


i^fi    ■z./^H 


COPYKIGHT,  1922,  BY  GINN  ANM)  COMPANY 

ENTERED  AT  STATIONERS'  HALL 

ALL   RIGHTS   RESERVED 

722.7 


Cbt   gtfttnKum   3Prt<* 

(;iNN  AND  COMPANY-  PRO- 
PRIETORS •  BOSTON  •  U.S.A. 


PREFACE 

This  work  has  been  prepared  for  the  purpose  of  providing 
a  thoroughly  usable  textbook  in  projective  geometry.  It  is  not 
intended  to  be  an  elaborate  scientific  treatise  on  the  subject, 
unfitted  to  classroom  use ;  neither  has  it  been  prepared  for 
the  purpose  of  setting  forth  any  special  method  of  treatment ; 
it  aims  at  presenting  the  leading  facts  of  the  subject  clearly, 
succinctly,  and  with  the  hope  of  furnishing  to  college  students 
an  interesting  approach  to  this  very  attractive  and  important 
branch  of  mathematics. 

There  are  at  least  three  classes  of  students  for  whom  a  study 
of  the  subject  is  unquestionably  desirable ;  namely,  those  who 
expect  to  proceed  to  the  domain  of  higher  mathematics,  those 
who  are  intending  to  take  degrees  in  engineering,  and  those 
who  look  forward  to  teaching  in  the  secondary  schools. 
Although  the  value  of  the  subject  to  the  second  of  these 
classes  has  not  as  yet  been  duly  recognized  in  America, 
European  teachers  for  several  decades  have  realized  its  useful- 
ness as  a  theoretical  basis  for  some  of  the  practical  work  in  this 
field.  For  the  large  number  of  students  belonging  to  the  third 
class,  trigonometry,  analytic  geometry,  and  projective  geometry 
are  the  three  subjects  essential  to  a  fair  knowledge  of  elemen- 
tary geometry,  and  it  is  believed  that  the  presentation  given 
in  this  book  is  such  as  greatly  to  aid  the  future  teacher.  There 
is  a  healthy  and  growing  feeling  in  America  that  teachers  of 
secondary  mathematics  need  a  more  thorough  training  in  the 
subject  matter,  even  at  the  expense  of  some  of  the  theory  of 
education  which  they  now  have.  This  being  the  case,  one  of 
the  best  fields  for  their  study  is  projective  geometry. 


iv  PREFACE 

It  is  recognized  that  students  of  projective  geometry  have 
usually  completed  an  elementary  course  in  analytic  geometry 
and  the  calculus,  that  they  have  a  taste  for  mathematics  which 
leads  them  to  elect  this  branch  of  the  science,  and  that  there- 
fore there  may  fittingly  be  some  departure  from  the  elementary 
methods  employed  in  the  earlier  mathematical  subjects.  On  the 
other  hand,  for  some  students  at  least,  projective  geometry  is  a 
transition  stage  to  higher  mathematics,  and  the  subject  should 
tlierefore  be  presented  with  due  attention  to  the  important 
and  recognized  principles  which  must  always  be  followed  in 
the  preparation  of  a  usable  textbook. 

It  is  the  belief  of  the  authors  that  they  have  followed  these 
principles  in  such  a  way  as  to  afford  to  college  students  a  simple 
but  sufficient  introduction  to  this  interesting  and  valuable 
branch  of  geometry.  Especial  attention  has  been  given  to  the 
proper  paging  of  the  book,  to  a  clear  presentation  of  the  great 
basal  propositions,  to  the  illustrations  accompanying  the  text, 
to  the  number  and  careful  grading  of  the  exercises,  and  to  the 
application  of  projective  geometry  to  the  more  elementary  field 
of  ordinary  Euclidean  geometry. 


CONTENTS 

PART  I.   GENERAL  THEORY 

CHAPTER  PAOB 

I.   Introduction 1 

11.   Principle  of  Duality 15 

III.  Metric  Relations.   Anharmonic  Ratio     ....  21 

IV.  Harmonic  Forms 31 

V.   Figures  in  Plane  Homology 37 

VI.   Projectivities  of  Prime  Forms 41 

VII.   Superposed  Projective  Forms 63 

PART  II.  APPLICATIONS 

VIII.   Projectively  Generated  Figures 79 

IX.   Figures  of  the  Second  Order 101 

X.   CoNics 115 

XI.     CONICS  AND   THE   ELEMENTS  AT   INFINITY        ....  143 

XII.   Poles  and  Polars  of  Conics 151 

XIII.  Quadric  Cones 167 

XIV.  Skew  Ruled  Surfaces       173 

HISTORY  OF  PROJECTIVE  GEOMETRY 181 

INDEX 185 


GREEK  ALPHABET 

The  use  of  letters  to  represent  both  iminbers  and  geometric 
magnitudes  has  become  so  extensive  in  mathematics  tjiat  it  is 
convenient  for  certain  purposes  to  employ  the  letters  of  the 
Greek  alphabet.  In  projective  geometry  the  Greek  letters  are 
used  particularly  to  represent  planes  and  angles.  These  letters 
with  their  names  are  as  follows : 


A 

n 

al[)ha 

N 

V 

nu 

B 

)8 

beta 

H 

i 

xi 

r 

y 

gamma 

0 

o 

omicron 

A 

8 

delta 

n 

TT 

l>i 

E 

£ 

ei)silon 

p 

P 

rho 

Z 

C 

/eta 

2 

0-,  s 

sigma 

H 

V 

eta 

T 

T 

tau 

© 

e 

theta 

Y 

V 

upsilon 

I 

I 

iota 

4> 

'i> 

phi 

K 

K 

kappa 

X 

X 

chi 

A 

X 

lambda 

* 

^ 

psi 

M 

H' 

mu 

n 

0) 

omega 

ELEMENTS  OF 
PROJECTIVE  GEOMETRY 

PART  I.    GENERAL  THEORY 

CHAPTER  I 

INTEODUCTIOK 

1.  Orthogonal  Projection.  In  elementary  geometry  the 
projection  of  a  point  upon  a  line  or  upon  a  plane  is  usually 
defined  as  the  foot  of  the  perpendicular  from  the  point  to 
the  line  or  to  the  plane,  and  the  projection  of  a  line  is  defined 
as  the  line  determined  by  the  projections  of  all  its  points. 
This  simple  projection  is  called  orthogonal  projection. 

oA 
I 


A  A\  A2 

Fig.  1  Fig.  2  Fig.  3 

Thus,  the  point  A'  in  Fig.  1  represents  the  orthogonal  projection 
of  the  point  A  upon  the  line  Z;  the  line  A{A2  in  Fig.  2  represents 
the  projection  of  the  line  A^A^  upon  the  line  I;  and  the  line  if  in 
Fig.  3  represents  the  projection  of  the  curve  k  upon  the  plane  a. 

2.  Symbols.  Projective  geometry,  like  other  branches 
of  mathematics,  employs  special  symbols,  generally  using 
capital  letters  to  denote  points,  small  letters  to  denote  lines, 
the  first  letters  of  the  Greek  alphabet,  a,  y8,  7,  S,  •  •  • ,  to  de- 
note planes,  and  the  Greek  letters  (f>  and  6  to  denote  angles. 

1 


2 


INTRODUCTION 


3.  Parallel  Projection.  In  a  plane  a  which  contains  a 
line  p  and  the  pomts  A^,  A^,  ^3,  •  •  •,  ^„,  if  a  line  I  is  drawn 
making  an  angle  <^  with  p,  each  of  the  lines  through 
^j,  ^3,  -rdg,  •  •  •,  An  parallel  to  I  makes  with  p  the  angle  0. 

A, 


a 

//Y 

%                         An 

A, 


"^'^ 


i- 


The  points  A\^  A'^,  A'^,  •  •  •,  A'^,  in  which  these  lines  inter- 
sect p,  are  called  the  projections  of  A^,  A^,  A^,  •  •  •,  A^  upon 
jt;  and  are  said  to  be  found  by  parallel  projection. 

In  space  of  three  dimensions  a  plane  figure  A^A^A^  may 
be  projected  upon  a  plane  ir  by  parallel  projection. 

4.  Central  Projection.  If  several  coplanar  points  ^j,  A^t 
A^,  •  •  •,  An  are  joined  to  a  point  P  of  their  plane,  and  if 
these  lines  are  cut  by  a  line  p,  the  points  of  intersection 


4 

AK^ 

-o4.". 

A, 

^ 

^^'2 

a; 

a 

of  the  lines  FA^,  PA^,  PA^,  . . .,  PA,,  with  the  line  p  are 
called  the  projections  of  ^j,  Jg^  ^3>  •  •  •»  ^n  from  the  center 
P  upon  ^  and  are  said  to  be  found  by  central  projection. 

In  space  of  three  dimensions  a  plane  figure  A^A^A^A^ 
may  be  projected  centrally  upon  a  plane  ir. 


KINDS  OF  PROJECTION  3 

5.  Projection  from  an  Axis.    Let  A^,  A^,  A^,  •  •  •,  A,^  be 

points  which  are  not  all  coplanar.  Their  orthogonal  pro- 
jections upon  a  Ime  p  may  be  obtamed  by  drawing  a  line 
from  each  of  them  perpendicular  to  p,  or 
by  passing  a  plane  through  each  of  them 
perpendicular  to  p.  The  latter  method 
may  be  generalized  by  requiring  merely 
that  the  planes  passed  through  the 
points  shall  be  parallel  to  a  fixed  plane 
which  is  not  necessarily  perpendicular 
to  p,  or  by  requiring  that  the  planes 
passed  through  the  points  shall  pass 
also  through  a  fixed  line  p'  instead  of  making  them  parallel 
to  a  fixed  plane,  p'  being  different  from  p.  Then  the  points 
are  said  to  be  projected  from  an  axis,  and  this  second  kmd  of 
projection  is  c&Wed  projection  from  an  aris  or  axial  projection. 

The  projection  of  points  by  parallel  planes  is  the  limiting  case  of 
projection  from  an  axis  in  which  the  axis  has  receded  indefinitely. 

6.  Operations  of  Projection  and  Section.  The  process  of 
finding  the  projection  of  a  plane  figure  upon  a  line  or 
plane  consists  of  two  parts.  The  first  part  is  called  the 
operation  of  projection  and  consists  in  the  construction  of  a 
figure  composed  of  lines,  or  of  planes,  or  of  both  Imes  and 
planes,  passing  through  the  points  and  lines  of  the  figure 
and  through  the  center  or  axis  of  projection.  These  lines 
and  planes  are  called  the  projectors  of  the  points  and 
lines  of  the  figure,  and  constitute  the  projector  of  the  figure. 

The  second  part  is  called  the  operation  of  section  and 
consists  in  cutting  the  projector  of  the  figure  by  a  line  or 
plane  called  the  line  of  projection  or  the  plane  of  projection. 

The  center  or  axis  of  projection,  and  the  line  or  plane  of  projection, 
should  be  so  taken  as  not  to  be  parts  of  the  figure  to  be  projected. 


4  INTRODUCTION 

Exercise  1.    Simple  Projections 

1.  Draw  a  figure  showing  the  orthogonal  projection  of  a 
given  circle  upon  a  given  plane. 

The  figure  may  be  drawn  freehand,  and  at  least  three  cases  should 
be  considered  :  (1)  the  circle  parallel  to  the  plane  ;  (2)  the  circle  oblique 
to  the  plane  ;  (3)  the  circle  perpendicular  to  the  plane.  In  (2)  consider 
also  the  case  in  which  the  circle  cuts  the  plane. 

2.  Draw  a  figure  showing  the  parallel,  but  not  necessarily 
orthogonal,  projection  of  a  given  square  upon  a  given  plane. 

3.  Draw  a  figure  showing  the  central  projection  of  a  given 
straight  line  or  a  given  plane  curve  upon  a  given  line. 

4.  Draw  a  figure  showing  the  central  projection  of  a  given 
square  upon  a  given  plane. 

Consider  three  cases,  as  in  Ex.  1.  Consider  the  case  in  which  P  13 
between  the  square  and  the  plane  as  well  as  the  case  in  which  it  is  not. 

5.  Draw  a  figure  in  which  the  four  vertices  of  a  square 
are  projected  from  a  given  axis  upon  a  given  line. 

In  projecting  from  a  center  P  upon  a  plane  tt,  describe  the 
projectors  and  the  projections  of  the  following,  mentionin(/  all 
the  noteworthy  special  cases  under  each : 

6.  A  set  of  points.  10.  Two  intersecting  lines. 

7.  A  line.  11.  Two  parallel  lines. 

8.  A  triangle.  12.  A  quadrilateral. 

9.  A  circle.  13.  A  pentagon. 

14.  Four  points  and  the  lines  joining  them  in  pairs. 

15.  A  tangent  to  a  circle  at  any  point. 

In  projecting  from  an  axis  p'  upon  (1)  a  plane  ir  and 
(2)  a  line  jo,  describe  the  projectors  and  the  projections  of  the 
following,  mentioning  the  noteworthy  special  cases  under  each : 

16.  A  set  of  points.  18.  A  line  not  parallel  to^'. 

17.  A  line  parallel  to  p'.  19.  A  circle. 


ELEMENTS  AT  INFINITY  5 

7.  Elements  at  Infinity.  Consideriiig  central  projection 
only,  and  supposing  the  center  P  and  the  plane  of  projec- 
tion TT  to  be  given,  these  questions  now  deserve  attention : 

1.  Does  every  point  of  a  plane  a  have  a  projector? 
Does  it  have  a  projection  ? 

Since  every  point  of  the  plane  a  can  be  joined  to  P  by  a  straight 
line,  every  point  of  a  has  a  projector;  but  since  this  projector  may 
happen  to  be  parallel  to  tt,  a  point  of  a  may  have  no  projection. 

2.  Is  every  line  which  passes  through  P  the  projector 
of  some  point  of  a  ? 

No ;  for  certain  of  these  lines  may  be  jiarallel  to  a. 

3.  Does  every  line  of  a  liave  a  projector  ?  Does  it  have 
a  projection  ? 

Consider  the  answers  to  Question  1.    Draw  the  figure. 

4.  Is  every  plane  which  passes  through  P  the  projector 
of  some  line  of  a  ? 

Consider  the  answer  to  Question  2.    Draw  the  figure. 

Certain  exceptional  cases  have  been  suggested  in  con- 
nection -with  these  questions.  Their  occurrence  is  due  to 
the  existence  of  parallel  lines  and  parallel  planes,  and  the 
difficulty  caused  by  them  may  be  removed  as  follows : 

Every  straight  line  is  assumed  to  have  one  and  only 
one  infinitely  distant  point,  and  this  point  is  called  the 
point  at  infinity  of  the  line. 

Every  plane  is  treated  as  having  one  and  only  one 
straight  line  situated  entirely  at  an  infinite  distance,  and 
as  having  all  its  infinitely  distant  pomts  situated  on  that 
line.    This  line  is  called  the  line  at  infinity  of  the  plane. 

Space  is  treated  as  having  one  and  only  one  plane  situ- 
ated entirely  at  an  infinite  distance,  and  as  having  all  its 
infinitely  distant  pomts  and  lines  situated  on  that  plane. 
This  plane  is  called  the  plane  at  infinity. 


a 

C^ 

c' 

Ci\  \ 
P    A, 

\^ 

A, 

A, 

6  INTRODUCTION 

8.  Illustrations.  Let  a  line  c  rotate  in  a  plane  a  about 
a  point  C  of  that  plane,  and  when  it  has  the  positions 
Cyy  c^"-')  let  it  meet  a  fixed  line  p  in  the  points  A^^ 
^2, . . ..  Then  as  long  as  c  is  not  parallel  to  p  it  meets 
p  in  one  point  and  only  one.  As  the  point  of  intersection 
becomes  more  and  more  dis- 
tant, the  line  c  becomes  more 
and  more  nearly  parallel  to 
p.  The  limiting  position  of  c 
as  the  point  of  intersection 
recedes  infinitely  is  the  line  c' 
through  C  parallel  to  p.  If, 
however,  the  rotation  of  c  is  continued  ever  so  little  beyond 
c\  the  intersection  of  c  and  p  is  found  to  be  at  a  great 
distance  in  the  other  direction  on  p,  and  as  the  rotation 
proceeds  farther  this  point  of  intersection  comes  continu- 
ously back  toward  Ay  Hence  c'  is  said  to  meet  jt?  in  a 
point  at  infinity.  If  there  were  several  infinitely  distant 
points  on  jt?,  they  would  with  C  determine  several  lines 
through  C  parallel  to  p,  or  several  of  these  points  would  be 
common  to  c'  and  jo,  or  one  or  more  of  these  points  taken 
with  C  would  fail  to  determine  a  straight  Ime.  Apparent 
conflict  with  propositions  of  Euclidean  geometry  is  best 
avoided  by  the  assumption  that  every  line  not  situated  at  an 
infinite  distance  has  one  and  only  one  infinitely  distant  point. 

Now  consider  all  points  of  a  plane  which  are  infinitely 
distant.  In  elementary  geometry  we  find  that  the  only 
plane  locus  met  by  every  line  of  its  plane  in  one  and  only 
one  point  is  a  straight  line.  The  locus  of  infinitely  distant 
points  of  the  plane  also  possesses  this  property.  Hence  this 
latter  locus  is  called  the  (straight)  line  at  infinity  of  the 
plane.  Similarly,  the  locus  of  the  infinitely  distant  points 
in  space  is  called  the  plane  at  infinity. 


PROJECTORS  AND  PROJECTIONS       7 

9.  Ideas  of  Projector  and  Projection  Simplified.  From  the 
considerations  set  forth  in  §§  7  and  8  it  appears  that  the 
introduction  of  the  elements  at  infinity  lias  distinct  advan- 
tages arising  out  of  the  fact  that,  from  the  new  point  of 
view,  statements  can  often  be  more  briefly  and  more  simply 
made.  The  greater  simplicity  is  due  to  the  fact  that  certain 
cases  involved  in  the  questions  cease  to  be  exceptional. 

For  example,  in  dealing  with  the  first  question  in  §  7  we  now 
say  that  every  point  of  a  plane  a  has  a  projection  upon  the  plane  tt  ; 
for  if  a  projector  happens  to  be  parallel  to  tt,  it  is  regarded  as  meet- 
ing TT  in  one  point  at  infinity. 

It  is  also  clear  that  we  may  now  say  that  all  the  lines  through  a 
point  P,  and  lying  in  a  plane  determined  by  P  and  a  line  a,  consti- 
tute the  jirojector  from  the  center  P  of  all  the  points  of  a. 

Similarly,  we  may  say  that  all  the  lines  and  all  the  planes  passing 
through  P  constitute  respectively  the  projectors  from  the  center  P 
of  all  the  points  and  all  the  lines  of  any  plane  not  passing  through  P; 
and  that  all  the  planes  through  any  line  p  form  the  projector  from 
the  axis  p  of  all  the  points  of  any  line  not  parallel  to  p. 

Exercise  2.    Projectors  and  Projections 

1.  Draw  figures  illustrating  the  four  statements  in  the  last 
three  paragraphs  above. 

2.  Consider  the  truth  of  the  statement  that  two  lines  in 
a  plane  have  one  and  only  one  common  point.  Illustrate  the 
statement  by  a  figure. 

3.  Do  every  two  planes  in  a  space  of  three  dimensions 
determine  a  line  ?    Explain  the  statement. 

4.  In  what  case  do  a  straight  line  and  a  plane  fail  to 
determine  within  a  finite  distance  exactly  one  point  ? 

5.  In  what  case  do  a  straight  line  and  a  point  fail  to 
determine  a  plane  ? 

6.  Two  straight  lines  which  determine  a  plane  determine, 
without  exception,  a  point. 


8  INTRODUCTION 

10.  The  Ten  Prime  Forms.  As  fundamental  sets  of  ele- 
ments we  use  the  following  sets,  called  the  ten  prime  forms  : 

One-Dimensional  Forms.  1.  The  totality  of  points  of  a 
straight  luie  (the  base')  is  called  a  range  of  points,  a  range, 
or,  less  frequently,  a  pencil  of  points. 

The  distinction  between  a  line  and  the  totality  of  its  points  may 
be  appreciated  by  considering  jioiuts  as  arranged  on  a  line  like  beads 
on  a  string.    Similar  considerations  apply  to  the  other  prime  forms. 

2.  The  totality  of  planes  through  a  straight  line  (the 
base')  is  called  an  axial  pencil. 

This  is  also  called  a  pencil  of  planes  or  a  slieaf  nf  planes. 

3.  The  totality  of  straight  lines  in  a  plane  and  through 
a  point  of  the  plane  is  called  a  flat  pencil. 

In  a  flat  j>encil,  either  the  point  common  to  the  lines  or  the  plane 
containing  the  lines  may  be  regarded  as  the  base  of  the  pencil. 

The  terms  range  of  points,  axial  pencil,  and  flat  pencil  are  used  for 
a  finite  number  as  well  as  for  an  infinite  number  of  elements. 

Two-Dimensional  Forms.  4.  The  totality  of  points  in  a 
plane  (the  base)  is  called  a  plane  of  points. 

5.  The  totality  of  planes  through  a  point  (the  base) 
is  called  a  bundle  of  planes. 

6.  The  totality  of  lines  in  a  plane  (the  base)  is  called 
a  plane  of  lines. 

7.  The  totality  of  lines  through  a  point  (the  base)  is 
called  a  bundle  of  lines. 

It  is  also  called  a  sheaf  of  lines,  but  because  the  word  sheaf  is 
used  in  conflicting  senses,  we  shall  not  use  it. 

Three-Dimensional  Forms.  8.  The  totality  of  points  of 
three-dimensional  space. 

9.  The  totality  of  planes  of  three-dimensional  space. 

Four -Dimensional  Form.  10.  The  totality  of  lines  of  three- 
dimensional  space. 


THE  TEX  PRIME  FORMS  9 

Exercise  3.    The  Ten  Prime  Forms 

Draw  a  rough  sketch  to  illustrate  each  of  the  following  : 

1.  Range  of  points.  4.  Plane  of  points. 

2.  Axial  pencil.  5.  Bundle  of  planes. 

3.  Flat  pencil.  6.  Plane  of  lines. 

Examine  each  of  the  following  prime  forms  when  the  base 
is  at  infiyiity : 

7.  Plat  pencil.  9.  Bundle  of  lines. 

8.  Axial  pencil.  10.  Bundle  of  planes. 

11.  Find  the  central  projection  of  a  range  of  points ;  of  a 
flat  pencil ;  of  a  plane  of  points  ;  of  a  plane  of  lines. 

12.  Find  the  plane  section  of  a  flat  pencil ;  of  an  axial 
pencil ;  of  a  bundle  of  planes  ;  of  a  bundle  of  lines. 

13.  Find  the  axial  projection  of  a  range  of  points  and  also 
of  a  flat  pencil,  the  axis  passing  through  the  base. 

14.  Find  the  linear  section  of  an  axial  pencil  and  also  of  a 
flat  pencil,  the  line  of  section  being  in  the  plane. 

15.  Investigate  the  central  projection  of  a  bundle  of  lines  ; 
of  the  points  of  space ;  of  the  lines  of  space. 

16.  Investigate  the  plane  section  of  a  plane  of  lines ;  of  the 
planes  of  space ;  of  the  lines  of  space. 

17.  Investigate  the  projection  from  an  axis  of  a  plane  of 
points  and  also  of  the  points  of  space. 

18.  Investigate  the  linear  sections  of  a  bundle  of  planes  and 
also  of  the  planes  of  space. 

19.  Apply  each  of  the  four  operations  to  the  prime  forms 
not  already  considered  in  connection  with  it. 

20.  Examine  the  results  of  Exs.  11-19  and  in  each  case 
determine  whether  to  every  element  of  the  original  figure  there 
corresponds  one  element  and  only  one  element  of  the  resulting 
figure,  and  vice  versa. 


10 


INTRODUCTION 


11.  Classification  of  Prime  Forms.  In  each  of  the  first 
three  classes  of  the  ten  prime  forms  mentioned  in  §  10 
the  prime  forms  of  every  possible  pair  are  connected  by 
a  simple  relation. 

Consider  first  a  range  of 
points  A^A^A^  >  >  •  A^  •  •  •  on  a 
base  j3,  and  consider  its  pro- 
jector a^a^a^  •••««•••  from  a 
pomt  P  exterior  to  /?,  this  pro- 
jector being  manifestly  a  flat  pencil.  By  setting  up,  or 
arranging,  the  infinitely  many  pairs  of  elements  A^,  a^; 
^2,  a^\  ^3,  a^',  •••;  A^^  a»;  •  •  •,  we  find  that  for  every 
point  of  the  range  there  is  a  corresponding  line  of  the 
flat  pencil,  and  vice  versa ;  and  that  if  two  points  are 
nearly  coincident,  so  also  are  the  corresponding  lines. 

Next,  make  a  section  of  an  axial  pencil  by  a  plane  tt. 
From  the  planes  a^,  a^^  a^,  •  •  •,  »„,  • .  •  of  the  axial  pencil 
and  the  lines  aj,  ag,  ag,  •  • .,  a,^,  •  • .  of  the  section  of  these 
planes  by  the  plane  tt,  infinitely 
many  pairs  of  elements  a^,  a^i 
a^,  aj;  ag,  a^;  •  •  •;  a,i,  a„;  •  •  •  may 
be  set  up.  Evidently  for  every 
plane  of  the  axial  pencil  there 
is  a  line  of  its  section  (the  flat 
pencil),  and  vice  versa ;  similarly 
for  the  range  and  the  axial  pencil. 

A  similar  conclusion  may  be 
reached  regarding  any  two  prime  forms  of  the  second  class, 
and  also  regarding  the  two  prime  forms  of  the  third 
class.  In  each  case,  by  the  setting  up  of  the  pairs,  there  is 
established  a  one-to-one  correspondence  between  the  elements 
of  the  two  forms. 

This  is  often  written  as  a  1  —  1  correspondence. 


PERSPECTIVITY 


11 


12.  Perspectivity.  Certain  cases  of  one-to-one  corre- 
spondence between  the  elements  of  prime  forms  of  the  same 
kind  should  also  be  noted.  For  example,  in  this  figure  if  two 
transversals  j9j,  jt?(  cut 
the  lines  a^,  a^,  -  •  >,  a,,, 

•  •  •  of    a  flat  pencil 
in  the  pomts  A^,  A[ ; 

A    A'  •  .  .  .'  4    A' '  . .  . 

the  ranges  A^A^-  •  • 
and  A\A',^  •  •  •  corre- 
spond in  this  way. 

Similarly,  in  this  figure,  if  two  flat  pencils  a^a^  •••«,, 


and  rtj«2 


a„  ' ' '  are  so  situated  that  a 


J,     Mj,       M2» 


ttn,  a, 


^H^  C'L\  '  '  '  intersect  in 
the  points  A^,  A^,  •  •  •, 
J„,  •  •  •  of  a  range,  such 
a  correspondence  exists. 

The  correspondences 
in  the  cases  in  §  11  re- 
sulted from  one  opera- 
tion of  projection  or  one  of  section.  In  the  cases  just  men- 
tioned the  correspondences  resulted  from  one  operation 
of  projection  and  one  of  section.  All  these  cases  and  other 
similar  cases  may  be  brought  into  one  group  by  means  of 
the  following  definition : 

If  either  of  two  prime  forms  can  be  obtained  from  the 
other  by  means  of  one  operation  of  projection,  or  one  opera- 
tion of  section,  or  by  means  of  one  operation  of  each  kind, 
the  two  forms  are  said  to  be  perspectively  related,  or  to  be 
in  perspective,  or  to  be  perspective. 

The  symbol  ^  is  often  used  for  "  is  perspective  with." 
The  perspective  relation  is  called  a  perspectivity. 


12  INTRODUCTION 

Exercise  4.    Perspectivity  and  Projection 

1.  If  the  line  a'  of  the  plane  a'  is  the  projection,  from  the 
center  P,  of  the  line  a  of  the  plane  a,  then  the  lines  a  and  a 
intersect  in  a  point  on  the  line  of  intersection  of  a  and  a'. 

2.  If  the  angle  formed  by  the  lines  a[  and  al  of  the  plane 
«'  is  the  ])rojection,  from  the  center  P,  of  the  angle  formed  by 
the  lines  a^  and  a.^  of  the  plane  a,  the  pairs  of  lines  a^,  a[ ; 
«2,  a.2  intersect  in  points  on  the  line  of  intersection  of  a  and  a'. 

3.  If  two  triangles  A^A^A^  and  A'^Al^A'^  of  the  planes  a  and 
«'  respectively  are  so  situated  that  the  lines  AiA[,  A^A'^,  and 
Af^A^  pass  through  a  common  point  P,  the  intersections  of 
the  pairs  of  sides  A^A^,  A'lA^;  A.^A^,  A'^A'^;  A^A^,  A'^A[  are 
collinear. 

4.  If  two  polygons  A^A^  -  •  •  A^  and  A'lA!^  •  •  •  Al^  of  the 
planes  a  and  «'  respectively  are  so  situated  that  the  lines 
A^A[,  ^2^2>  •  •  •>  ^M^«  P^ss  through  a  common  point  P,  the 
intersections  of  the  pairs  of  sides  A^A^,  A^A'^ ;  A^A^,  Al^A'^;  •  •  -; 
yl„/1j,  ^,',^1,  and  the  intersections  of  the  pairs  of  diagonals 
'^I'^s'  ^"^i^^si  A^A^,  .'I1-I4;  •  •  • ;  A^A^,  A^Ai]  •  •  • ;  ^^.-l„j,  ^yL-'^M?  *  '  ' 
are  collinear. 

It  will  be  noticed  that  Exs.  1-4  form  a  related  set  of  problems,  as  is 
also  the  case  with  Exs.  6-8. 

5.  If  two  lines  a  and  a'  of  the  planes  a  and  a'  respectively 
intersect,  either  may  be  regarded  as  the  projection  of  the  other 
from  any  point  exterior  to  both  lines  but  in  their  common  plane. 

6.  State  and  prove  the  converse  of  Ex.  2. 

7.  State  and  prove  the  converse  of  Ex.  3. 

8.  State  and  prove  the  converse  of  Ex.  4. 

9.  Given  three  points  on  a  line  a  and  a  point  A^  not  on  the 
line,  construct  a  triangle  that  shall  have  vlj  as  a  vertex  and 
shall  have  each  of  its  sides,  produced  if  necessary,  pass  through 
one  and  only  one  of  the  three  given  points.  How  many  such 
triangles  can  be  constructed  ? 


PERSPECTIVITY  AND  PROJECTION  13 

10.  Investigate  the  problem  similar  to  Ex.  9  in  which  two 
given  points  A^  and  A^  of  the  plane  a  are  to  be  vertices  of  the 
required  triangle,  and  show  how  to  construct  the  triangle  when 
such  a  triangle  exists. 

11.  Given  the  points  Ai  and  A[  of  two  planes  a  and  a'  which 
intersect  in  a  given  line  a,  and  given  in  the  plane  a  a  triangle 
constructed  as  required  in  Ex.  9,  use  Ex.  3  to  obtain  a  triangle 
in  the  plane  a'  that  shall  have  ^{  as  a  vertex  and  shall  have 
sides  which,  produced  if  necessary,  shall  intersect  the  line  a 
in  the  points  in  which  this  line  is  cut  by  the  sides  of  the  given 
triangle  in  a.    How  many  constructions  are  possible  ? 

12.  Investigate  the  cases  of  Ex.  11  in  which  a  second  vertex 
of  one  or  of  each  of  the  triangles  is  also  given. 

13.  If  two  triangles  AiA.2A^  and  A'^Al^A^  in  the  same  plane 
a  are  so  situated  that  the  lines  A^A'^,  A^A[,  and  A^Al^  are  con- 
current, the  intersections  oiA^A^,  ^1^2  >  ^2^3? ^2^3  >  -^gJi,  A'^A[ 
are  collinear. 

Let  A^A^  and  A\A'^  meet  in  Cj,  A^A^  and  A'^A'^  in  C■^,  and  A^A^ 
and  A'^A\  in  Cg.  Take  a  center  of  projection  P  not  in  tlie  plane  a,  and 
project  the  wliole  figure  upon  a  plane  parallel  to  the  plane  PC^C^,  thus 
obtaining  the  line  at  infinity  as  the  projection  of  CgC^.  Prove  that  the 
projection  of  Co  is  on  this  line. 

The  development  of  this  problem  and  similar  problems  is  fully 
considered  in  Chapter  V. 

14.  State  and  prove  the  converse  of  Ex.  13. 

15.  State  and  prove  the  proposition  of  plane  geometry  which 
corresponds  to  Ex.  4. 

16.  State  and  prove  the  converse  of  Ex.  15. 

17.  Given  three  points  A^,  A^  A^  on  a  line  a  in  a  plane  a, 
and  three  points  A[,  A 2,  A'^  on  a  line  a'also  in  the  plane  a,  find 
three  points  A[',  .4",  Ag,  not  necessarily  collineav,  into  which 
both  sets  of  three  points  can  be  projected. 

18.  With  the  same  data  as  in  Ex.  17  find  three  collinear 
points  A[',  A!/,  A'^'  into  which  the  first  two  sets  of  three  points 
mentioned  can  be  projected. 


14  INTRODUCTION 

19.  In  Ex.  17  consider  also  the  case  in  wliicli  the  lines  a 
and  a!  are  coincident  and  in  which  the  points  A[,  Al,  A'^  are 
not  necessarily  all  distinct  from  the  points  .1,,  A,^,  A^. 

20.  Given  three  points  on  a  line  a,  construct  a  quadrilateral 
such  that  the  pairs  of  opposite  sides  shall  intersect  in  two  of 
the  given  points,  and  such  that  one  of  its  diagonals  shall  pass 
through  the  other  point. 

21.  Assuming  the  construction  asked  for  in  Ex.  20,  use  Ex.  4 
and  Ex.  16  to  obtain  additional  quadrilaterals  fulfilling  the 
same  conditions  as  the  first.  Do  the  other  diagonals  of  these 
quadrilaterals  intersect  ? 

22.  Given  two  quadrilaterals  so  constructed  as  to  fulfill  the 
conditions  of  Ex.  20,  the  straight  lines  joining  corresponding 
vertices  of  these  figures  are  concurrent. 

23.  If  the  quadrilateral  constructed  in  Ex.  20  moves  so  as 
continuously  to  fulfill  the  conditions  stated,  the  other  diagonal 
constantly  passes  through  a  fixed  point. 

24.  Show  how  to  find  the  fixed  point  mentioned  in  Ex.  23. 

25.  In  Ex.  20,  if  the  third  of  the  given  points  bisects 
the  segment  joining  the  other  two  given  points,  determine  the 
position  of  the  fixed  point  mentioned  in  Ex.  23. 

26.  Given  five  points  on  a  line  a,  construct  a  quadrilateral 
such  that  each  of  its  sides  and  one  of  its  diagonals,  pro- 
duced if  necessary,  shall  pass  through  one  and  only  one  of 
the  given  points.  Obtain  additional  quadrilaterals  fulfilling 
the  same  conditions. 

27.  In  Ex.  2G  investigate  the  relation  that  the  other  diago- 
nals of  any  two  of  the  quadrilaterals  bear  to  each  other  and  to 
the  given  line  a. 

28.  Extend  the  problem  in  Ex.  26  to  the  case  of  the  pentagon. 

29.  Given  two  quadrilaterals  so  constructed  as  to  fulfill  the 
conditions  of  Ex.  26,  the  straight  lines  joining  pairs  of  corre- 
sponding vertices  of  these  figures  are  concurrent. 


CHAPTER  II 

PRINCIPLE  OF  DUALITY 

13.  Principle  of  Duality.  It  is  now  liighly  desirable  to 
consider  a  certain  important  relation  between  pairs  of 
figures  in  space,  and  also  between  their  properties.  The 
nature  of  this  relation,  by  the  use  of  which  the  difficulties 
of  the  subject  may  be  reduced  by  almost  half,  is  explained 
by  the  Principle  of  Dnalitt/,  or  the  Principle  of  Reciprocity, 
which  may  be  stated  as  follows: 

Corresponding  to  any  figure  in  space  which  is  made  vp  of 
or  generated  hy  points,  lines,  and  planes  there  exists  a  second 
figure  which  is  made  up  of  or  generated  hy  planes,  lines,  and 
points,  such  that  to  every  point,  every  line,  and  every  plane  of 
the  first  figure  there  corresponds  respectively  a  plane,  a  line, 
and  a  point  of  the  second  figure,  and  such  that  to  every  propo- 
sition which  relates  to  points,  lines,  and  planes  of  the  first 
figure,  but  which  does  not  essentially  involve  ideas  of  measure- 
ment, there  corresponds  a  similar  jyroposition  regarding  the 
planes,  lines,  and  points  of  the  second  figure,  and  these  two 
propositions  are  either  both  true  or  both  false. 

The  two  figures  which  are  related  in  the  manner  just 
described,  as  well  as  the  two  propositions,  are  said  to  be 
dual,  reciprocal,  or  correlative. 

As  a  simple  illustration  of  the  principle,  consider  the  following : 

Two  points  determine  a  line.  Two  planes  determine  a  line. 

Two  lines  through  a  point  deter-      Two  lines  in  a  plane  determine 
mine  a  plane.  a  point. 

15 


16  PRINCIPLE  OF  DUALITY 

14.  Assumption  of  the  Principle  of  Duality.  The  validity 
of  the  principle  of  duality  will  not  be  proved  in  this  book, 
although  it  is  possible  so  to  formulate  the  axioms  of  pro- 
jective geometry  that  they  are  unchanged  if  everywhere  the 
words  point  and  plane  are  interchanged,  and  thus  to  show 
this  validity.  Nevertheless  the  principle  will  be  applied 
Avith  great  frequency  in  deriving  properties  of  figures,  and 
in  so  doing  either  of  two  courses  may  be  adopted:  On 
the  one  hand,  it  may  be  assumed  that  the  prmciple  is  valid 
and  is  capable  of  a  proof  which  is,  of  course,  entirely  inde- 
pendent of  any  results  obtained  by  means  of  the  principle 
itself;  on  the  other  hand,  the  principle  may  be  used 
simply  as  the  basis  of  a  rule  for  formulating  the  dual  of 
any  proposition,  the  rule  being  justified  in  every  case  by  a 
proof  of  this  dual  proposition. 

Of  these  two  courses  the  latter  is  not  a  difficult  one,  for 
after  the  principle  of  duality  has  been  used  to  derive  the 
enunciation  of  the  dual  proposition,  it  may  be  applied  to 
the  various  steps  of  the  proof  of  the  original  proposition  to 
obtain  a  new  set  of  statements  which  may  be  examined 
to  see  if  they  constitute  a  proof  of  the  dual.  In  each  case  it 
will  be  found  that  a  proof  is  secured.  The  plan  has  the 
further  advantage  that  it  avoids  the  feeling  of  dissatisfac- 
tion and  uncertainty  attendant  upon  making  a  very  general 
and  far-reaching  but  apparently  unjustified  assumption, 
besides  which  the  repeated  application  of  the  principle 
leads  to  that  confidence  in  its  validity  which  comes  from 
increasing  experimental  evidence. 

For  this  reason  the  second  course  has  been  adopted  in 
this  book ;  and  whenever  the  dual  of  a  proposition  is  derived 
by  applying  the  principle  of  duality,  either  the  proof  of 
the  dual  is  derived  by  the  same  means  or  such  derivation 
is  left  to  serve  as  an  exercise  for  the  student. 


DERIVATION  OF  DUAL  PROPOSITIONS         17 

15.  Derivation  of  Dual  Propositions.  If  a  figure  or  a 
proposition  in  the  geometry  of  space  is  given,  the  first 
considerations  which  enter  into  the  derivation  of  the  dual 
figures  or  propositions  are  the  facts  that  the  point  and  the 
plane  are  dual  elements,  and  that  every  geometric  figure 
may  be  obtained  by  using  either  the  point  or  the  plane 
as  the  primary  generating  element.  Although  neither  the 
point  nor  the  plane  has  a  superior  claim  over  the  other 
to  be  considered  as  the  primary  generating  element,  it  fre- 
quently happens  that  the  statement  of  a  proposition  is  so 
framed  as  to  imply  that  one  or  the  other  of  these  elements 
has  been  so  used.  For  this  reason  the  derivation  of  the  dual 
of  any  proposition  generally  requires  more  than  the  mere 
interchange  of  the  words  point  and  plane.  On  the  other 
hand,  it  is  true,  almost  without  exception,  that  propositions 
which  have  duals  may  be  so  stated  that  the  latter  may  be 
found  from  the  former  by  such  interchange. 

Skill  in  the  derivation  of  dual  figures  and  propositions 
is  quickly  gained,  and  the  examples  given  in  §  16  will 
assist  the  beginner  in  acquiring  this  skill.  In  general  it 
may  be  said  that  the  student  will  find  it  advantageous  to 
consider,  in  every  proposition  he  meets,  the  proposition 
which  results  from  the  method  of  treatment  mentioned 
above,  and  then  to  consider  whether  the  proof  of  the 
derived  proposition  can  be  obtained  from  the  original 
proof  in  the  same  way. 

In  plane  geometry  the  pomt  and  the  line  are  dual  ele- 
ments, and  any  figure  may  be  regarded  as  having  been  gen- 
erated by  either  of  these  elements.  In  a  general  way,  duals 
in  the  plane  are  derived  by  interchanging  these  elements. 

The  principle  of  duality  for  threefold  space,  applied  to 
plane  geometry,  yields  the  geometry  of  the  bundle  of  lines 
and  planes,  the  line  and  the  plane  being  dual  elements. 


18 


PKINCIPLE  OF  DUALITY 


16.  Examples  of  Duality, 
the  more  simple  examples  of 

1.  Point  A. 

2.  Line  a. 

3.  Two  points  determine  a  line. 

4.  Two  lines  which  determine 

a  plane  also  determine  a 
point. 

5.  Three  points  in  general  de- 

termine a  plane. 

6.  Several  points  which  lie  in  a 

plane. 

7.  Several  lines  which  lie  in  a 

plane. 

8.  A    plane    triangle;    that    is, 

three  points  and  the  three 
lines  determined  by  them 
in  pairs. 

9.  A  plane  poh/gon. 

10.  A  range  of  points. 

11.  K  flat  pencil. 

12.  A  plane  of  points  and  a  plane 

of  lines. 

13.  Four  points  in  a  plane  and 

the  six  lines  joining  them 
in  pairs ;  a  coni])lete  quad- 
rangle, or  four-point. 

14.  Four  lines  in  a  plane  and  the 

six  points  determined  by 
the  various  pairs  of  these 
lines ;  a  comi)lete  quadri- 
lateral, or  four-line. 

15.  Four  points  in  space  and  the 

lines  and  planes  deter- 
mined by  them. 


The  followmg  are  a  few  of 
duality : 

1'.  Plane  a. 

2'.  Line  a'. 

3'.  Two  planes  determine  a  line. 

4'.  Two  litres  which  determine 

a  point  also  determine  a 

j)lane. 

5'.  Three  planes  in  general  de- 
termine a  point. 

6'.  Several  planes  which  pass 
through  a  point. 

T.  Several  lines  which  pass 
through   a  point. 

8'.  A  trihedral  angle;  that  is, 
three  planes  and  the  three 
lines  determined  by  them 
in  pairs. 

9'.  A  polyhedral  angle. 

10'.  An  axial  pencil. 

11'.  A  flat  pencil. 

12'.  A  bundle  of  planes  and  a 
bundle  of  lines. 

13'.  Four  planes  through  a  point 
and  the  six  lines  of  inter- 
section in  pairs;  a  com- 
plete four-flat. 

14'.  Four  lines  through  a  point 
and  the  six  planes  deter- 
mined by  the  various  pairs 
of  these  lines ;  a  complete 
four-edge. 

15'.  Four  planes  in  space  and 
the  lines  and  points  deter- 
mined by  them. 


EXAMPLES  OF  DUALITY 


19 


16.  Given   three   collinear  points 

A,  B,  C,  find  iovLT points  P^, 
P^,  P3,  P^  such  that  the 
•  lines  P^P^  and  P^P^  shall 
meet  in  A,  the  lines  Pg^a 
and  P4P1  shall  meet  in  B, 
and  the  lineP^P^  shall  pass 
through  C. 

17.  //  (he  line  a'  of  the  plane  a' 

is  the  projection  from  the 
center  P  of  a  line  a  of 
the  plane  a,  the  lines  a  and 
a'  intersect  in  a  point  of 
the  line  of  intersection  of  a 
and  a. 

Proof.  The  lines  a  and  a'  lie  in 
the  plane  determined  by  P  and 
a,  and  hence  they  determine  a 
point.  Since  their  jwint  of  in- 
tersection lies  in  the  jilane  a 
and  also  in  the  plane  a',  it 
lies  in  the  line  determined  by 
a  and  a'. 


16'.  Given  three  coaxial  planes  a, 
/3,  y,  find  four  planes  tTj, 
iTo,  TTj,  TT^  such  that  the 
lines  TTjTTg  and  Tr^TTi  shall 
lie  in  a,  the  lines  tt^tt^ 
and  TT^TTi  shall  lie  in  /?, 
and  the  line  tt^tt^  shall  lie 
in  y. 

17'.  If  the  line  a'  through  the  point 
A'  is  determined  by  A'  and 
the  point  of  intersection  of 
the  plane  it  tcitk  the  line  a 
through  the  point  A,  the  lines 
a  and  a'  lie  in  a  plane  which 
passes  through  the  line  A  A', 

Proof.  The  lines  a  and  a'  pass 
through  the  point  determined 
by  7r  and  a,  and  hence  they 
determine  a  plane.  Since  their 
jilane  passes  through  the  point 
^1  and  also  through  the  point 
A',  it  passes  through  the  line 
determined  by  A  andyl'. 


17.  Figures  in  a  Plane  and  Figures  in  a  Bundle.    The 

Principle  of  Duality  may  also  be  stated  for  the  geometry 
of  figures  in  a  plane,  and  likewise  for  the  geometry  of 
figures  in  a  bundle.  In  the  first  case  the  point  and  the  line 
are  dual  elements,  and  in  the  second  case  the  line  and 
the  plane.  Simple  modifications  of  the  statement  of  the 
principle  in  §13  yield  the  statements  for  the  two  cases. 

Pairs  of  the  examples  of  §  16  may  be  used  to  illustrate  this  fact. 
For  example,  the  following  pairs  are  duals  in  the  plane :  1,  2 ;  6,  7 ; 
8,  8;  9,9;  10,  11;  13,  14. 

The  following  pairs  are  duals  in  the  bundle  :  1',  2' ;  6',  7' ;  8',  8' ; 
9',  9';  10',  11';   13',  14'. 

Many  other  examples  of  duality  will  be  found  as  we  proceed. 


20  PRINCIPLE  OF  DUALITY 

Exercise  5.    Principle  of  Duality 

1.  By  means  of  the  Principle  of  Duality  obtain  for  three- 
fold space  the  statement  and  also  the  proof  of  the  dual  of  Ex.  3, 
I>age  12. 

2.  Similarly,  find  the  space  dual  of  Ex.  4,  page  12. 

3.  Derive  and  prove  the  space  dual  of  Ex.  13,  page  13. 

4.  Obtain  for  plane  geometry  the  statement  and  proof  of 
the  dual  of  Ex.  13,  page  13. 

5.  Derive  the  space  dual  of  Ex.  4. 

6.  Verify  that  in  the  geometry  of  the  bundle  the  results 
of  Exs.  3  and  5  are  dual. 

7.  If  a  proposition  in  plane  geometry  or  in  the  geom- 
etry of  the  bundle  has  a  dual,  but  is  not  self-dual  in  that 
geometry,  then  in  the  geometry  of  threefold  space  it  belongs 
to  a  set  of  four  propositions  each  of  which  is  dual  with  two 
of  the  others. 

8.  If  a  proposition  is  self-dual  in  plane  geometry,  then  the 
four  propositions  mentioned  in  Ex.  7  reduce  to  two. 

9.  Can  the  four  propositions  of  Ex.  7  ever  reduce  to  one 
proposition  ?  Discuss  in  full. 

10.  Give  an  example  of  a  self-dual  figure  in  threefold  space. 

11.  State  a  simple  self-dual  proposition  regarding  the  figure 
mentioned  in  the  answer  to  Ex.  10. 

12.  Are  all  propositions  regarding  the  self-dual  figure  of 
Ex.  10  themselves  self-dual  ?    Discuss  in  full. 

13.  Given  three  planes  passing  through  a  line  a  of  a  bundle 
whose  base  is  A,  construct  in  this  bundle  a  four-edge  such  that 
pairs  of  opposite  edges  lie  in  two  of  the  given  planes  and  one 
of  its  diagonal  lines  lies  in  the  remaining  plane. 

Compare  this  example  with  Ex.  20,  page  14.  Notice  that  when  a  prop- 
osition is  harder  to  prove  than  its  dual  the  proof  of  the  latter  may  be 
used  to  suggest  that  of  the  former. 


CHAPTER  III 

METRIC  RELATIONS.    ANHARMONIC  RATIO 

18.  Metric  and  Descriptive  Properties.  Properties  of  geo- 
metric figures  are  of  two  sorts  :  (1)  metric,  that  is,  those 
which  relate  to  the  measurement  of  geometric  magnitudes ; 
(2)  descriptive,  that  is,  those  which  are  not  metric.  Nearly 
all  the  propositions  of  ordinary  elementary  geometry  deal 
^vith  metric  properties,  while,  speaking  generally,  those  of 
projective  geometry  deal  with  descriptive  properties.'  In 
fact,  it  is  possible  to  exclude  almost  entirely  from  projec- 
tive geometry  the  consideration  of  the  metric  properties  of 
figures.  On  the  other  hand,  even  when  the  object  in  view 
is  the  study  of  the  descriptive  properties  of  figures,  it 
frequently  happens  that  brevity  is  secured  by  the  use  of 
metric  considerations.  For  this  reason  the  metric  proper- 
ties of  figures  will  be  used  freely  whenever  the  nature  of 
the  work  is  such  as  to  make  this  course  advisable. 

If  the  student  will  consider  the  work  which  has  thus  far  been  done 
in  this  book  he  will  see  that  no  statement  has  been  made  that  depends 
in  any  way  upon  measurement.  The  lines  projected  may  be  of  any 
desired  length,  the  angles  may  have  any  desired  measure,  and  the 
closed  figures  may  have  any  desired  area.  It  is  therefore  evident  that 
the  work  thus  far  has  not  been  metric  in  any  way. 

In  this  chapter,  on  the  other  hand,  we  proceed  to  establish  certain 
important  properties  which  will  prove  to  be  of  great  service  to  us  in 
subsequent  work.  IMoreover,  as  we  proceed,  it  will  appear  that  by 
virtue  of  the  propositions  proved  in  §§26  and  27  the  study  of  these 
properties  is  appropriate  in  connection  with  the  descriptive  properties 
of  figures. 

21 


22  METRIC  llELATIONS 

19.  Relations  of  Line  Segments.  In  measuring  distances 
along  a  straight  line  attention  is  given  to  direction  as  well 
as  to  length.  One  direction  along  a  line  is  selected  as  posi- 
tive, the  opposite  one  being  negative.  The  direction  of  a 
line  segment  is  called  its  sense  and  is  indicated  by  the 
order  of  the  end  letters,  JJi  denoting  the  segment  of  a  line 
thought  of  as  extendhig  from  A  to  B  and  BA  the  segment 
of  a  line  thought  of  as  extending  from  B  to  A.  Evidently, 
therefore,  we  have 

AB  =  -BA, 

or  AB+BA  =  0. 

Having  adopted  this  convention  with  respect  to  signs, 
many  identical  relations  can  be  proved.  For  example,  A, 
B,  Cy  •  •  'y  J,  K  being  coUmear  pomts  in  any  order : 

O 

O  O O— O  O         0  I  O  0 o— 

AD^B  CK    '  E         J 

1.  AB+BC+CD+..-  +  JK  +  KA  =  (i. 

2.  AB.CD+BC'AD  +  CA  •  BD  =  0. 

3.  BC  '  AD^+CA  .  in?  +  AB  .  ClP'  +  BC  -  CA  .  AB  =  0. 

In  one  method  of  proving  these  relations  we  employ 
as  origin  any  point  0  on  the  given  line.  Then  for  any 
segment  AB  we  have 

AB  =  OB-OA. 

This  substitution  and  others  of  a  similar  nature  being 
made  in  any  identity  of  this  sort,  the  truth  of  the  identity 
becomes  apparent. 

The  proof  may  be  made  algebraic  if  the  measures  of  the 
segments  OAj  OB,  •  • .  are  denoted  by  the  letters  a,  5,  •  •  •. 
Moreover,  having  an  identical  relation  among  real  alge- 
braic numbers,  we  may  deduce  a  corresponding  relation 
among  line  segments. 


LINE  SEGMENTS  AND  ANGLES  23 

20.  Relations  of  Angles.  In  measuring  angles  attention 
is  given  to  tlie  direction  of  rotation  as  well  as  to  the  magni- 
tude of  the  angles.  Rotation  is  called  positive  if  it  proceeds 
in  the  direction  opposite  to  that  taken  by  the  hands  of  a 
clock ;  otherwise  it  is  called  negative.  The  direction  of 
rotation  is  called  the  sense  of  the  angle  and  is  mdicated  by 
the  order  of  the  letters  which  denote  its  arms,  the  angle 
formed  by  the  rotation  of  a  line  from  the  position  of  the 
line  a  to  that  of  the  line  b  being  called  the  angle  ab. 

The  ambiguity  which  may  be  felt  to  attach  to  this 
method  of  representing  angles  may  easily  be  removed  in 
the  following  way :  Take  as  the  standard  line  any  line  o 
through  the  intersection  of  the  lines  a  and  b,  and  let  it  be 
agreed  that  the  angle  between  a  and  b 
shall  be  understood  to  mean  that  angle 
formed  by  the  lines  a  and  b  which  does 
not  contain  the  line  o,  and  that  the  angle 
oa  formed  by  o  and  a  shall  mean  the 
angle  included  between  a  specified  one 
of  the  halves  of  the  infinite  line  o  which  proceed  from  the 
point  common  to  a  and  b  and  that  half  of  a  which  is  first 
reached  by  a  positive  rotation  from  o. 

Then  the  algebraic  identities  by  means  of  which  the  rela- 
tions between  line  segments  were  proved,  as  well  as  all 
other  algebraic  identities,  are  capable  of  interpretations  with 
respect  to  angles.  In  the  above  case  ab  —  ob  —  oa,  and  by 
means  of  such  identities  the  relations  may  be  verified. 

The  dihedral  angles  formed  by  pairs  of  planes  of  an 
axial  pencil  may  be  treated  in  a  similar  fashion,  a  standard 
plane  (o  being  used.  In  this  case  the  angle  between  two 
planes  a  and  ^  will  be  denoted  by  a^  if  the  planes  a,  /3, 
and  o)  have  the  same  general  positions  as  the  lines  a,  5, 
and  0  in  the  figure  shown  above. 


24  ANHARMONIC  RATIO 

21.  Anharmonic  Ratio.  The  most  useful  metric  element 
in  projective  geometry  is  called  an  anharmonic  ratio.  It  is 
related  to  a  range  of  four  points,  a  flat  pencil  of  four  lines, 
and  an  axial  pencil  of  four  planes,  as  follows : 

1.  The  anharmonic  ratio  (^ABCD^  of  four  collinear  points 
A,  B,  C,  D,  is  defined  as     .^      .  ^ 

'bc'Iw' 

2.  TJie  anharmonic  ratio  (ahcd^  of  four  concurrent  and 
coplanar  line  segments  a,  J,  c,  d  is  defined  as 

sin  ac  ^  sin  ad 
sin  he    sin  hd 

3.  The  anharmonic  ratio  (a^yB^  of  four  coaxial  planes  a,  ^, 

7,  8  is  defined  as  •  •       s 

''  ''  sm  ay  ^  sm  ad 

sin  /Sy  *  sin  /38 

An  anharmonic  ratio  is  also  called  a  cross  ratio  or  a  double  ratio. 
The  anharmonic  ratio  (ABCD)  is  easily  remembered  by  writing 

A       A 

— — :  — —  and  then  writing  C  in  both  terms  of  the  first  fraction  and 

D  in  both  terms  of  the  second  fraction. 

The  above  definition  of  anharmonic  ratio,  though  not  universal, 
has  the  approval  of  the  leading  authorities  of  the  present  time. 

22.  CoKOLLAKY.   If  A,  B,  C,  D  are  collinear  points^  then: 

1.  (^ABCD^  is  negative  when  and  only  when  the  segment 
AB  contains  either  C  or  Z),  hut  not  hoth. 

2.  (^ABCU)  approaches  AC/BG  as  a  limit  as  D  recedes 
indefinitely  in  either  direction. 

Exercise  6.    Anharmonic  Ratios 

1.  If  (ABCD^)  =  (ABCD^),  D^  and  D^  are  coincident. 

2.  Consider  Ex.  1  and  §  22  for  the  anharmonic  ratio  (nhed). 

3.  Consider  Ex,  1  and  §  22  for  the  anharmonic  ratio  (n^yB). 


RATIOS  OF  FOUR  POINTS  25 

23.  Twenty-four  Anharmonic  Ratios.  Corresponding  to 
the  order  A,  B,  C,  D  of  four  collinear  points,  there  has  been 
defined  the  anharmonic  ratio  (^ABCD^.  There  are,  however, 
twenty-four  possible  orders  for  these  points,  that  is,  the  4 ! 
permutations  of  the  four  letters ;  and  therefore  there  are 
twenty -four  anharmonic  ratios  for  the  four  points,  as  follows: 

QABCD),  {ABDC),  QACBD),  (ACDB),  (ADBC),  (ABCB'), 
(BACD},  (BABC),  (BCAD),  (BCDA},  (^BBAC),  {BDCA\ 
(CABD),  (^CADB),  (CBAB),  (CBDA^,  {CDAB^,  {CBBA^, 
(BABC),  (^DACB),  (BBAC),  (^DBCA),  (DCAB),  (DCBA). 
But  by  definition  (§  21) 

^  -^     AC   AD      AC'BD 

while  (^ABI>C)  =  ^:^=^^^, 

^  ^     BI)    BC     AC'BD 

and  so  •  {BACD^  =  {ABDC). 

In  like  manner  it  may  be  shown  that  the  last  eighteen 
ratios  fall  into  six  sets  of  three  each,  all  those  in  any  set 
being  equal  to  one  of  the  first  six  anharmonic  ratios. 

Exercise  7.    Anharmonic  Ratios 
Given  the  three  collinear  points  A,  B,  C,  proceed  as  follows : 

1.  Find  the  collinear  point  D  such  that  {ABCD)=^  7. 

2.  Find  the  collinear  point  D  such  that  (ABCD)  =  —  7. 

3.  Find  the  collinear  point  D  such  that  (ABCD)=  k. 

4.  Prove  that  (ABCD)  =  (BADC)  =  (CDAB)  =  (DCBA). 

5.  Determine  the  several  pernmtations  of  the  four  elements 
A,  B,  C,  D  which  leave  the  value  of  (ABCD)  unchanged. 

6.  Which  of  the  twenty -four  ratios  are  equal  to  (A  BBC)  ? 


26  ANHARMONIC  RATIO 

24.  Relations  of  the  First  Six  Ratios.  The  first  six 
anharmonic  ratios  given  in  §  23  are  also  connected  by  sim- 
ple relations.    If  (^ABCn)=x,  we  have  the  following: 

1.  (ABCD)=x. 

rAnnns      ^^    ^^      AC-BD 
lor  ^ABCD)  =  —  :—  =  j^-^  =  x, 

^  ^      BD    BC      AC'BD      X 

3.   (^ACBD^  =  \-x. 

For  (§  19,  2)  AB  ■  CD  +  BC •  AD  +  CA  ■  BD^  0; 

AB    CD     ,       CA-BD     ,       AC  AD     , 
^•'^"•^^  AD'CB^^-ADTcB^^-BC-Bb^^-''' 

Therefore     (J  CBD)  =  \-x. 
1 


4.  (ACDB}  = 


1- 


a; 


For  (yl  CDB)  = — -— ,  since  we  have  simply  interchanged  the 

'^  ^      (ACBD)  ^-^  ° 

last  two  letters,  as  in  1  and  2  above.    Hence  the  result  follows  from  3. 

5.  (ADBC}  =  ^^' 

X 

For  (^  CBD)  =  \  —  X,  by  3,  where  we   merely  interchange  the 
second  and  third  letters.    Hence,  by  similar  reasoning, 

{A  DBC)  =  1  -  (^  BDC)  =  1-1  =  ini . 

6.  (ADCB)=^-- 

x  —  1 

For  we  found  from  1  and  2  that  the  transposition  of  the  third  and 
fourth  letters  gave  the  reciprocal  of  the  original  anharmonic  ratio, 

1  X 

and  so  from  5  we  have  (ADCB)  = = • 

^  ^      (A  DBC)      x-1 


RELATIONS  OF  THE  RATIOS 


27 


25.  Equality  of  the  Six  Expressions.     We  may  now  de- 
termine the  values  of  x  for  which  any  pair  of  the  six 

1     ^  1         x—\  ,       X  , 

expressions  x^-t  \  —  x^ ,  ,  and are  equal, 

X  1  —  X  X  x  —  1 

and  therefore  we  may  determine  the  values  of  these  six 
expressions  which  correspond  to  the  values  of  x  so 
found.    The  results  may  be  put  in  tabular  form  as  follows : 


X 

1 

X 

1-x 

1 

1-X 

x-1 

X 

X 

x-1 

1 

1 

0 

CO 

0 

oc 

-1 

-1 

2 

h 

2 

h 

h 

l+V-3 

2 
l-V-3 

l-V-3 

2 
l+V-3 

-1 

l+V-3 

-1 

l-V-3 

2 

2 

2 

2 

2 

2 

l_V-3 

l+V-3 

l+V-3 

l-V-3 

l-V-3 

l  +  V-3 

2 

2 

2 

2 

2 

2 

0 

GO 

1 

1 

X 

0 

2 

i 

-1 

-1 

i 

2 

CO 

0 

X! 

0 

1 

1 

It  will  be  noticed  that  these  values  of  the  six  distinct 
ratios  of  the  4 !  anharmonic  ratios  may  be  classified  into 
three  groups  as  follows : 

1.  Those  in  which  the  values  of  x  are  imaginary,  the 
values  of  all  the  functions  being  also  imaginary. 

2.  Those  in  which  the  values  of  x  are  1,  0,  or  oo,  the 
values  of  the  functions  being  also  1,  0,  or  oo. 

3.  Those  in  which  the  values  of  x  are  —1,  ^,  or  2,  the 
values  of  the  functions  being  also  —1,  ^,  or  2. 

The  first  group  is  not  concerned  with  the  anharmonic  ratio  of 
four  real  collinear  points,  and  the  second  does  not  correspond  to  the 
anharmonic  ratio  of  four  real  points  which  are  distinct.  The  third 
group  is  the  only  important  one  for  our  present  purpose,  and  this 
will  be  considered  in  Chapter  IV. 

PO 


28 


ANHARMONIC  RATIO 


Theorem.  Prime  Forms  Related  by  projections 
AND  Sections 

26.  If  two  prime  forms  of  the  first  class  are  so  related,  that 
either  may  he  obtained  from  the  other  by  a  finite  number  of 
projections  and  sections,  the  anharmonic  ratio  of  any  four 
elements  of  one  is  equal  to  the  anharmonic  ratio  of  the  corre- 
sponding four  elements  of  the  other. 

Proof.  I.  Let  either  form  be  obtainable  from  the  other 
by  means  of  one  operation  of  projection  or  one  of  section. 

1.  Range  ABCD  and  fiat  pencil  abed. 

From  P,  the  base  of  the  flat  pencil,  draw  PQ  perpendicular 
to  jP,  the  base  of  the  range  ABCD.  Then,  equating  pairs 
of  expressions  for  double  the 
areas  of  the  triangles  A  CP,  BCPy 
ADP,  BDP,  we  have 

PA'PCs,\nac  =  PQ.AC, 

PB.  PC  sin  be  =:PQ.BC, 

PA  .  PD  sin  ad^PQ'  AD, 

PB  .  PD  sin  bd  =  PQ'  BD. 

A  C      PA    sin  ad 
liC'     PB 


TT           P^    sin  ac 
Hence    • 


and 
whence 


PB    sin  be 

sin  ac    sin  ad 


sin  bd 
AC  AD, 
BC'  BD' 


AD 

bd' 


sin  be    sin  bd 

(abed)  =  (ABCD). 
2.  Mange  ABCD  and  axial  pencil  a^yS. 

From  a  point  P  in  the  base  of  the  axial  pencil  a^yS  project 
the  range  ABCD,  obtaining  as  projector  the  flat  pencil  abed. 

Then  (ABCD)  =  (abcd).  But  in  Case  3  it  will  be  shown 
that  (abcd)=(a^yh).    Hence  (ABCD)=  (a^yh). 


EELATIONS  OF  THE  RATIOS 


29 


3.  Flat  pencil  ahcd  and  axial  pencil  a^y8. 

Through  a  point  ^  in  the  base  of  the  axial  pencil  a/378 
pass  a  plane  perpendicular  to  this  base,  cutting  the  planes 
a,  y8,  7,  S  in  the  lines  aQ,  6q,  Cq,  d^  and  the  lines  a,  b,  c,  d  in 
the  points  A,  B,  C,  D.  From  the 
definition  of  the  angles  between  the 
planes  it  follows  that 

But    (aQhQCf^dQ)  =  {ABCn) 

= (abcd^. 
Hence    {a^yS  )  =  (ahcd^ . 

II.  Let  either  form  be  obtainable  from  the  other  by 
means  of  several  operations  of  projection  or  of  section, 
or  of  both. 

Let  two  prime  forms  /^  and  /„  +  i  be  obtainable,  either 
from  the  other,  by  means  of  n  operations,  and  let  the  prime 
forms  which  are  successively  produced  beginning  with  f^ 
t»e  /2,/3,  •  •  ',fu,fn  +  v  Also  let  e^,  ej,,  4',  ej,"  and  e^+i,  4  +  r 
4+p  ^i+i  be  corresponding  sets  of  four  elements  of  two 
consecutive  prime  forms /;fc,/;t+i. 

Then,  by  Part  I,     (e^el^e^'e^")  =  (e^  + 14  + 14'+ 14'+ 1)- 

This  relation  is  true  for  all  values  of  k  from  1  to  n. 

Hence         (^i^iWW")  =  (^n+i««+i«n+i««+i)» 
and  the  truth  of  the  theorem  is  established. 

27.  Corollary.  If  two  prime  forms  of  the  first  class  are 
perspective,  the  anharmonic  ratio  of  any  four  elements  of 
one  form  is  equal  to  that  of  the  four  corresponding  elements 
of  the  other  form. 


30  ANHARMONIC  RATIO 

Exercise  8.    Relations  of  the  Ratios 

1.  Show  how  the  table  on  page  27  is  obtained,  verifying 
each  result. 

2.  Any  two  letters  in  the  anharmonic  ratio  (A BCD)  may 
be  interchanged  without  affecting  the  value  of  the  ratio,  pro- 
vided the  other  two  letters  are  also  interchanged. 

T^7.  .  1       ^  1  x  —  1  X 

In  the  8ix  expressions  x,  — »  1—  x,   '  »  

X  1  —  X  X  x  —  1 

make  successively  the  following  substitutions  for  x  and  note  the 
recurrence  of  the  original  forms  of  the  expressions: 

x'  —  1 

3.  x'.  5.  1-x'.  7.  ^-r^- 

x' 

4.  -:•  6.  :: :•  8. 


x'  1-x'  x'-l 

9.  If  the  point  0  and  three  nonconcurrent  lines  a,  b,  c  are 
in  one  plane,  draw  a  line  through  O  which  shall  cut  a,  b,  c  in 
points  A,  B,  C  such  that  (OABC)=  k,  any  given  number. 

10.  Solve  the  dual  of  Ex.  9  in  plane  geometry. 

11.  Solve  the  space  dual  of  Ex.  9. 

12.  If  A^,  A^,  /Jj,  B^,  Cj,  Cj,  Z)j,  Dg,  X,  Y  are  collinear  points, 
and  if  (A^A^XY)  =  {B^B^XY)  =  (C^C^XY)  =  (D^D.^XY)  =  -1,  it 
follows  that  (A^B^C^D^)  =  (A^B^C^D^). 

13.  li  A^,  A^,  ■  • ',  A„,X,  Y  are  w  +  2  collinear  points,  then 
(A^A^XY)(A^A^XY)  •  •  •  (A,,_,A^XY)(A^A,XY)=1. 

14.  If  ylj,  A^,  A^,  X,  Y  are  five  coplanar  points,  and  if 
A^(A^A^XY)  denotes  the  anharmonic  ratio  of  the  four  lines 
from  A^  to  A^,  A^,  X,  Y,  it  follows  that  the  product  of  the 
anharmonic  ratios  A^(A^A^XY),  A^(A^A^XY),  A^(A^A.^XY)  is  1. 

15.  Generalize  the  result  found  in  Ex.  14. 

16.  If  /I J,  A^  A^,  A^,  X,  Fare  concyclic  points,  it  follows  that 
the  anharmonic  ratios  A'(Jj/l2^l8^^)and  Y(A^A^A^A^)  are  equal. 


CHAPTER  IV 
HARMONIC  FORMS 

28.  Harmonic  Range.  When  four  collinear  points  A,  B, 
C,  D  are  so  situated  that  {ABCD)  =  —1,  the  four  points 
are  said  to  constitute  a  harmonic  range. 

In  the  same  way  we  may  define  a  harmonic  flat  pencil  and 
a  harmonic  axial  pencil. 

Any  one  of  these  three  forms  is  spoken  of  as  a  harmonic 
form.,  and  it  follows  (§  26}  that  every  form  derived  from  a 
harmonic  form  by  a  finite  number  of  projections  and  sec- 
tions is  a  harmonic  form.  If  three  elements  of  a  harmonic 
form  are  given,  it  is  evident  that  the  fourth  element,  called 
the  fourth  harmonic  to  the  three,  is  uniquely  determined. 

Moreover,  since  the  anharmonic  ratio  is  here  negative, 
from  the  above  definition  the  elements  of  the  first  pair, 
say  A  and  B.,  separate  those  of  the  second  pair,  say  C  and  D. 

Ti.  +  •       •         /^P^m  1    ACBD  ,         .AC  AD 

That  IS,  since  (ABCD)  =  —  1, -•  =  —  1,  and = , 

'^  ^  'AD-BC  BC  BD 

so  that  the  two  ratios  have  opposite  signs.  Therefore,  either  Cor  D 

divides  A  B  internally  and  the  other  divides  it  externally. 

The  elements  of  either  of  these  pairs,  A  and  B,  or  C 
and  D,  are  said  to  be  conjugates  or  harmonic  conjugates 
with  respect  to  the  other  pair.  They  are  also  said  to  be 
harmonically  separated  by  the  elements  of  the  other  pair. 

Since  from  the  definition  (§  21)  the  anharmonic  ratios  (ABCD) 
and  (CDAB)  are  equal,  it  follovv^s  that  the  relation  betw^een  the  pairs 
of  elements  A,  B  and  C,  D  is  symmetric  with  respect  to  them. 

31 


32  HARMONIC  FORMS 

Exercise  9.   Harmonic  Ranges 

1.  Given  three  collinear  points  A,  B,  C,  with  C  bisecting 
AB,  determine  the  fourth  harmonic  D. 

2.  Consider  Ex.  1  when  C  is  the  point  at  infinity. 

3.  Consider  Ex.  1  when  A,  B,  C  are  any  collinear  points. 

4.  Given  three  concurrent  lines  a,  h,  c,  with  c  bisecting  the 
angle  ab,  determine  the  fourth  harmonic  d. 

Compare  Ex.  4  with  Ex.  1. 

5.  Consider  Ex.  4  when  a,  h,  c  are  any  concurrent  lines. 
Compare  Ex.  5  with  Ex.  3. 

6.  If  {ABCD)  =  —  1,  the  four  points  A,  B,  C,  D  may,  by  a 
finite  number  of  projections  and  sections,  be  projected  into  the 
positions  A,  B,  D,  C. 

7.  If  A  BCD  is  a  harmonic  range,  the  line  segments  AC, 
AB,  and  AD  are  connected  by  the  proportion  AC:AD  = 
AC  —  AB:AB  —  AD. 

8.  If  AB^Cjy^  and  AB^C^D^  are  harmonic  ranges  on  differ- 
ent bases,  the  lines  B^B^,  ^i^a?  -^1^2  ^^®  concurrent,  and  the 
lines  B^B^,  ^1^2'  ^-Pi  ^^'^  ^^^  concurrent. 

9.  If  AJi^C^D^  and  Aji^CJ)^  are  harmonic  ranges,  and  if 
^j/lj,  i^i^a,  CjCg  are  concurrent  at  0,  then  D^^  also  passes 
through  0. 

10.  If  A,  B,  C,  D,  0,  P  are  points  on  a  circle  and  are 
so  placed  that  the  pencil  O(ABCD)  is  harmonic,  the  pencil 
P  (A  BCD')  is  also  harmonic. 

11.  If  ABCD  is  a  harmonic  range,  and  if  O  is  the  midpoint 
of  CD,  then  ()C^  =  OA  ■  OB. 

12.  Use  the  result  of  Ex.  11  to  find  a  pair  of  points  which 
shall  be  harmonic  conjugates  with  respect  to  two  given  pairs 
of  collinear  points  A^,  B^;  A^,  B^. 

13.  Given  four  coplanar  lines,  draw  when  possible  a  line 
which  shall  cut  them  in  a  harmonic  range. 


COMPLETE  QUADRANGLE  33 

29.  Complete  Quadrangle.  The  figure  formed  by  four 
points  in  a  plane,  no  three  of  which  are  colhnear  (asP,  Q,  E,  S 
in  the  figure  below),  and  the  six  lines  determined  by  them 
is  called  a  complete  four-point  or  complete  quadrangle. 

Any  two  of  the  six  lines  of  a  complete  quadrangle  which 
do  not  intersect  in  one  of  the  original  four  points  are  called 
opposite  sides.  The  intersections  of  opposite  sides  are  called 
diagonal  points,  and  they  are  the  vertices  of  the  diagonal 
triangle  of  the  complete  quadrangle. 

Theorem,  harmonic  Property  of  a  Quadrangle 

30.  If  four  collinear  points  A,  B,  C,  D  are  so  situated  that 
two  opposite  sides  of  a  complete  quadrangle  pass  through  A, 
two  opposite  sides  piass  th'ough  B,  and  the  two  remaining  sides 
pass  through  C  and  D  respectively,  then  (^ABCD)  —  —  1. 


Let  P,  Q,  R,  S  be  the  vertices  of  the  complete  quadrangle, 
and  let  PQ,  BS  pass  through  A ;  PS,  QR  through  B ;  PR 
through  C;  and  QS  through  D. 

Then       {ABCD^  =  iSQOD)  =  {BACD^  =  (JWcm  ' 

Hence  (ABCI>y  =  l, 

and  {ABCI)}  =  -1. 

(A BCD)  cannot  be  equal  to  +  1,  since  no  two  points  are  coinci- 
dent, as  would  then  be  the  case. 


34 


HARMONIC  FORMS 


31.  Complete  Quadrilateral.  The  figure  formed  by  four 
lines  in  a  plane,  no  three  of  which  are  concurrent  (as 
JO,  q,  r,  8  below),  and  the  six  points  determined  by  them  is 
called  a  complete  four-side  or  complete  quadrilateral. 

Any  two  of  the  six  pomts  of  a  complete  quadrilateral 
which  do  not  both  lie  on  one  of  the  original  four  lines 
are  called  opposite  vertices.  The  lines  determined  by  pairs 
of  opposite  vertices  are  called 
diagonal  lines,  and  they  deter- 
mine the  diagonal  triangle. 

The  student  should  compare  this  fig- 
ure with  that  of  the  complete  quadrangle 
in  §  30,  and  should  notice  also  the  duality 
suggested  by  §§29  and  31,  the  dual  ele- 
ments being  the  point  and  line. 


Theorem.  Harmonic  Property  of  a  Quadrilateral 

32.  Tf  four  concurrent  lines  a,  b,  c,  d  are  so  situated  that 
two  opposite  vertices  of  a  complete  quadrilateral  are  on  a,  two 
opposite  vertices  on  b,  and  the  two  remaining  vertices  on  c  and 
d  respectively,  then  (abcd^  =  —  1. 

Let  p,  q,  r,  s  in  the  figure  above  be  the  sides  of  the 
complete  quadrilateral,  and  let  p  and  q,  and  also  r  and  s, 
intersect  on  a ;  p  and  «,  and  also  q  and  r,  intersect  on  b ; 
p  and  r  intersect  on  c ;  and  s  and  q  intersect  on  d. 

(abed)  =  (sqod)  =  (bacd)  = 


Then 
Hence 
and 


(abcdy  =  1, 
(abed)  =  —  1. 


(abed) 


Why  cannot  (abed)  =  -f- 1  ?    Students  should  compare  this  proof, 
step  by  step,  with  that  of  §  30. 


QUADRANGLES  AND  QUADRILATERALS        35 

Exercise  10.    Quadrangles  and  Quadrilaterals 

1.  State  and  prove  the  converse  of  §  30. 

2.  State  and  prove  the  converse  of  §  32. 

3.  Two  vertices  of  the  diagonal  triangle  of  a  complete 
quadrangle  are  harmonically  separated  by  the  points  in  which 
the  line  determined  by  them  is  cut  by  the  remaining  pair  of 
opposite  sides  of  the  quadrangle. 

4.  By  interchanging  certain  elements,  it  is  possible  to 
derive  §  32  from  §  30  and  Ex.  2  above  from  Ex.  1.  Derive  a 
proposition  in  this  way  from  Ex.  3  and  investigate  its  truth. 

5.  From  §  30  derive  a  theorem  respecting  the  complete 
four-flat  and  prove  it. 

In  the  geometry  of  space  the  figure  dual  to  the  complete  (luadraugle 
is  called  the  complete  four-fiat,  and,  similarly,  the  complete  four-edge  iu 
Ex.  6  is  dual  to  the  complete  quadrilateral. 

6.  As  in  Ex.  5,  from  §  32  derive  a  theorem  respecting  the 
complete  four-edge  and  prove  it. 

7.  The  six  points,  other  than. the  diagonal  points,  in  which 
the  diagonal  lines  meet  the  sides  of  a  complete  quadrangle 
lie  in  sets  of  three  on  each  of  four  lines. 

8.  From  the  result  in  Ex.  7  prove  the  existence  of  a  com- 
plete quadrilateral  which  has  the  same  diagonal  triangle  as  any 
complete  quadrangle. 

9.  Prove  the  plane  duals  of  Exs.  7  and  8. 

10.  In  this  figure  QS  is  parallel  to  AB.    Show 
that  PC  is  a  median  and  is  divided  harmonically. 

Consider  D,  the  intersection  of  QS  and  AB,  to  have 
moved  to  infinity. 

11.  In  the  complete  quadrangle  shown  in  §  30  show  that 
AQ  ■  PS  -  BC  =  -  AC  ■  BS  .  PQ. 

12.  As  in  Ex.  11,  show  that  A  Q-  PS  .  BD  =  AD-  BS  •  PQ. 

13.  Using  §  30,  prove  Ex.  12,  page  30. 


36  HARMONIC  FOKMS 

33.  Descriptive  Definitions  of  Harmonic  Forms.  The  har- 
monic forms  might  originally  have  been  defined  in  a  purely 
descriptive  fashion  based  upon  the  facts  just  developed. 
Thus,  a  harmonic  range  might  have  been  defined  as  a  set 
of  four  collinear  points  so  situated  that  through  each  of 
the  first  two  points  there  pass  two  opposite  sides  of  a  com- 
plete quadrangle,  and  through  each  of  the  other  two  points 
there  passes  one  of  the  remaming  sides  of  the  quadrangle. 
Similar  definitions  might  have  been  given  for  the  har- 
monic flat  pencil  and  the  harmonic  axial  pencil.  These 
are  the  definitions  which  are  usually  adopted  when  it  is 
desired  to  avoid  as  far  as  may  be  possible  the  use  of  con- 
siderations based  upon  measurement. 

Exercise  11.   Harmonic  Forms 

1.  Given  three  collinear  points  A,  B,  C,  construct  the  fourth 
harmonic  D  from  the  descriptive  definition  of  §  33. 

In  the  constructions  on  this  page  use  only  an  ungraduated  ruler. 

2.  Given  three  concurrent  lines  a,  b,  c,  construct  the  fourth 
harmonic  d  from  the  descriptive  definition  of  §  33. 

3.  Given  a  line  segment  J 5  and  an  indefinite  line  parallel 
to  A  B,  bisect  ,1 B. 

4.  Given  a  line  segment  AB,  its  midpoint  C,  and  any  point 
0  not  in  the  line  of  .4/i,  through  O  draw  a  line  parallel  to  AB. 

5.  Given  two  intersecting  lines  and  the  bisector  of  one  of 
the  angles  formed  by  them,  construct  the  bisector  of  the 
supplementary  angle  formed  by  the  lines. 

6.  Given  a  line  segment  AB  divided  at  C  in  the  ratio  ??i :  n, 
construct  a  point  D  that  divides  the  segment  AB  externally 
in  the  same  ratio. 

7.  Dualize  for  space  the  descriptive  definitions  of  a  harmonic 
range  and  a  harmonic  flat  jxjncil. 


CHAPTER  V 


FIGURES  IN  PLANE  HOMOLOGY 


34.  Homologic  Plane  Figures.  Further  interesting  appli- 
cations of  the  anharmonic  ratio  and  illustrations  of  its 
significance  occur  in  homologic  plane  figures. 

Given  two  figures  in  a  plane,  if  to  every  point  of  one 
figure  there  corresponds  a  pomt  of  the  other,  if  to  every 
line  of  one  there  corre- 
sponds a  line  of  the  other, 
if  the  lines  joining  corre- 
sponding points  of  the 
two  figures  are  concur- 
rent, and  if  the  inter- 
sections of  corresponding 
lines  are  collinear,  the 
two  figures  are  said  to  be 
homologic,  or  in  (^plane) 
homology. 

The  point  in  wliich  all  lines  joining  corresponding  points 
are  concurrent  is  called  the  center  of  homology,  the  line 
which  contains  all  intersections  of  corresponding  lines  is 
called  the  axis  of  homology. 

In  the  above  illustration  the  two  given  figures  are  the  triangles 
ABC,  A'B'C  The  corresponding  points  indicated  are  the  three 
pairs  of  vertices,  but  any  number  of  other  pairs  of  points  may  be 
chosen.  The  corresponding  lines  are  AB  and  A'B',  BC  and  B'C, 
CA  and  C'A'.  The  center  of  homology  is  O,  and  the  axis  of 
homology  is  o. 

37 


38  FIGURES  IN  PLANE  HOMOLOGY 

Theorem.  Figures  in  homology 

35.  If  Uvo  figures  are  in  plane  homology^  it  follows  that : 

1.  All  sets  of  four  collinear  points  consistitig  of  the  center  of 
homology,  a  point  on  the  axis  of  homology,  and  two  correspond- 
ing points  of  the  figure  have  a  common  anharmonic  ratio. 

2.  All  sets  of  four  concurrent  lines  consisting  of  a  line 
through  the  center  of  homology,  the  axis  of  homology,  arid  two 
corresponding  lines  of  the  figure  have  a  common  anharmonic 
ratio. 

3.  These  two  common  anliar77ionic  ratios  are  equal. 


Proof.  Let  A^,  A^,  A^  and  A[,  A^,  A^  be  corresponding  sets 
of  three  points  of  two  homologic  figures;  and  let  OA^A[, 
OA^A'^,  OA^A^  intersect  o  in  0^,  0^,  Og  respectively ;  also 
let  A^A^,  A[A'^  meet  in  i|;  A^A^,  A^A^  meet  in  i^ ;  and 
A^A^,  A'^A[  meet  in  I^.  Then  it  is  evident  that 
(00,A,A[}  =  iO(\A^A!,~)  =  (00,A,A^} 


CONSTANT  OF  HOMOLOGY  39 

36.  Constant  of  Homology.  The  common  value  of  the 
two  anharmonic  ratios  found  in  the  theorem  of  §  35  is  called 
the  constant  of  homology  for  the  two  figures. 

Given  a  center  0  and  an  axis  of  homology  o,  we  can 
construct  a  figure  homologic  with  any  given  figure  in  such 
a  way  that  to  any  point  A  of  the  given  figure  there  shall 
correspond  any  selected  point  A'  on  the  line  OA,  or  that  to 
any  line  a  of  the  given  figure  there  shall  correspond  any 
selected  line  a'  concurrent  with  o  and  a.  For  the  point  A' 
selected  there  is  a  value  of  the  constant  of  homology. 
Conversely,  for  any  assigned  value  of  the  constant  of 
homology,  the  center  and  axis  being  given,  one  and  only 
one  point  A'  corresponds  to  any  given  point  ^  of  a  given 
figure.  The  fact  is  that  tvhen  the  center,  axis,  and  constant 
of  homology/  are  given,  one  and  only  one  figure  homologic  with 
a  given  figure  can  he  constructed. 

Examples  of  such  constructions  are  given  on  page  40. 

There  are  several  notable  special  cases  of  the  homologic 
relation.  One  such  case  is  that  of  harmonic  homology  in 
which  the  constant  of  homology  is  —  1. 

Another  case  is  that  in  which  the  axis  of  homology  is 
the  line  at  infinity.  Then  all  pairs  of  corresponding  lines 
are  parallel,  and  the  figures  are  similar.  In  this  case  the 
constant  of  homology  (^OO-^A^A'^  becomes  OA^/OA[,  which 
may  be  shown  to  be  the  ratio  of  similitude.  If  in  addition 
the  constant  is  —1,  the  center  0  is  a  center  of  symmetry 
for  the  figure  composed  of  the  two  homologic  figures. 

A  third  case  is  that  in  which  the  center  of  homology  is 
a  point  at  infinity.  Then  the  constant  is  O^A[/0-^^Ay  If 
also  the  homology  is  harmonic,  the  axis  of  homology  is 
an  axis  of  symmetry  for  the  figure  composed  of  the  two 
homologic  figures. 


40  FIGURES  IN  PLANE  HOMOLOGY 

Exercise  12.   Figures  in  Homology 

1.  Given  a  center  of  homology  O,  an  axis  of  homology  o, 
and  any  triangle  A^A^A^,  construct,  homologic  with  A^A^A^,  a 
triangle  that  has  a  vertex  ^^  at  a  given  position  on  OA^. 

As  to  the  possibility  of  this  construction,  see  Ex.  13,  page  13. 

Given  a  triangle  A^^A^A^^  the  center  of  homology  0,  the  axis 
of  homology  o,  and  the  following  constants  of  homology^  con- 
struct the  figures  homologic  with  A^A^A^  : 

2.  3.  3.  -3.  4.  -1.  5.  1. 

6.  In  each  of  Exs.  2-5,  if  A[A!^A'^  is  the  result  of  the  con- 
struction, use  the  same  center,  axis,  and  constant  of  homology 
to  construct  the  figure  homologic  with  A[A!^Al. 

7.  If  there  are  three  coplanar  figures  f,  /„  yj,  and  if  for 
a  given  center  and  a  given  axis  of  homology  two  of  them, 
/j  and  /jj,  are  homologic  with  the  constant  Cg,  and  if  f^  and  f^ 
are  homologic  with  the  constant  c^,  then  f^  and  f^  are  homo- 
logic  with  the  constant  of  homology  c^  equal  to  c^  •  c^. 

8.  Consider  carefully  Exs.  2-7  for  the  case  in  which  O  is 
a  point  at  infinity  and  the  case  in  which  a  is  the  line  at  infinity. 

9.  For  a  given  center,  axis,  and  constant  of  homology  con- 
struct the  line  corresponding  to  the  line  at  infinity. 

This  line  is  called  tlie  vanishing  line. 

10.  In  Exs.  2-5  determine  the  vanishing  line. 

11.  Under  what  conditions  is  the  vanishing  line  at  infinity? 

Given  a  circle,  the  axis  of  homology  o,  and  the  center  of 
Jiomology  0,  construct  the  figure  homologic  with  the  circle 
under  each  of  the  following  conditions : 

12.  The  vanishing  line  does  not  meet  the  circle. 

13.  The  vanishing  line  is  a  secant  of  the  circle. 

14.  The  vanishing  line  is  a  tangent  to  the  circle. 


CHAPTER  VI 
PROJECTIVITIES  OF  PEIME  FORMS 

37.  Projective  One-Dimensional  Prime  Forms.  Whenever 
there  exists  between  the  elements  of  two  one-dimensional 
prime  forms  a  one-to-one  correspondence  such  that,  by 
means  of  a  finite  number  of  operations  of  projection  and 
section,  it  is  possible  to  pass  simultaneously  from  all  the 
elements  of  one  prime  form  to  the  corresponding  elements 
of  the  other,  the  two  prime  forms  are  said  to  be  projectively 
related  or  to  be  projective. 

The  correspondence  existing  between  two  projective  one- 
dimensional  prime  forms  is  called  a  projectivity. 

The  symbol  j-  is  frequently  used  for  "  is  projective  with." 

Theorem.  Projective  Prime  Forms 

38.  Prime  forms  which  are  projective  with  the  same  prime 
form  are  projective  with  each  other. 

Proof.  If  Wj  operations  yield  a  form  f^  from  a  form  /j, 
and  if  n^  operations  yield /g  from/g,  then  n^  +  n^  operations 
yield  /g  from  f^ 

It  is  not  the  purpose  of  this  book  to  discuss  the  projectivities 
of  prime  forms  other  than  one-dimensional  ones,  but  it  may  be 
stated  that  between  the  prime  forms  of  higher  dimensions  there 
exist  relations  which  have  the  same  general  character  as  those 
just  defined,  and  that  these  relations  are  also  called  projectivities. 
A  complete  study  of  projective  geometry  would  include  the  consider- 
ation of  these  higher  projectivities  and  of  many  important  geometric 
propositions  relating  to  them. 

41 


42  PROJECTIVITIES  OF  PRIME  FORMS 

Theorem.  Projectivity  of  Triads 

39.  Between  tivo  one-dimensional  prime  forms,  each  of 
which  consists  of  three  elements  in  a  specified  order,  there 
exists  a  projectiinty. 


Proof.  If  either  of  the  prime  forms  is  not  a  range,  it  is 
possible  by  operations  of  projection  and  section  to  obtain 
from  it  a  range  which  is  projective  with  it.  Hence  it  is 
necessary  to  prove  the  proposition  only  for  the  case  in 
which  both  prime  forms  are  ranges  of  three  points. 

This  theorem  is  what  was  formerly  called  a  lemma,  a  proposition 
inserted  merely  for  the  purpose  of  leading  up  to  a  fundamental  theo- 
rem ;  in  this  case,  the  one  given  in  §  40.  The  proof  involves  the 
consideration  of  the  three  cases  below. 

1.   The  ranges  may  he  coplanar  and  upon  different  bases. 

Let  the  ranges  be  ^j^jCj  on  the  base  jt>j  and  ^2^2 ^2  ^^ 
the  beise  p^,  and  let  both  ranges  be  in  tlie  same  plane. 

Draw  the  line  through  A^  and  A^,  and  on  it  take  any 
points  I^  and  J^,  not  coincident  with  A^  and  Ac^  respectively. 

Draw  ^5j,  ^Cj,  J^B^,  J^C^;  and  let  P^B^,  J^B^  intersect 
at  B,  and  let  J^C^,  I^C^  intersect  at  C. 

Through  B  and  C  draw  the  line  p,  cutting  A-^A^  at  A. 

Then       range  A^B^C^  —  range  ^BC  —  range  A^B^C^. 
Hence  range  A^B^C^  —  range  A^B^C^.  §  38 


PROJECTIVITY  OF  TRIADS  43 

2.   The  ranges  may  he  coplanar  and  upon  the  same  base. 
Let  the  ranges  A^B^C^  and  ^2-^2^2  ^^  ^^  ^^^  same  base  p. 

~-^ — ■ 0  '■        a p 


^^A,  ^2     B,  cr^ 

0  ^  CI  n  '         0  ^  P 

B2  C2      ^2 

It  is  here  assumed  that  the  points  A^  B^  C^  may  be  the 
same  except  for  order  as  the  points  A^^  B^,  Cj,  or  may  be 
partly  or  wholly  distinct  from  these  points. 

In  any  case  from  a  center  P,  exterior  to  the  base  p, 
project  A^B^C^  upon  a  new  base  p'.  Then  apply  Case  1 
to  show  that  the  range  so  obtained  on  the  base  p'  is  pro- 
jective with  the  range  A^B^Cy  It  then  follows  that  the 
range  ^2^2 ^2  ^^  projective  with  the  range  A^B^Cy 


3.   TJie  ranges  may  not  he  coplanar. 


Let  the  bases  p-^^  and  p^  of  the  ranges  A-^B-^C^  and  ^2-^2 ^2 
not  be  in  any  one  plane. 

Join  any  point  Oj  of  the  base  p^  to  any  point  0^  of  the 
base  jt>2  by  the  line  p.    Select  three  points  A^  B,  C  on  p. 

Then,  by  Case  1,  it  follows  that 

range  ^2^2  ^2  a  ^^^^^  ABC^ 
and  range  A^B^C-^  —  range  ABC. 

Therefore       range  ^2^2 ^2  a  i"^"g6  A^B^Cy 
From  these  results  the  existence  of  a  projectivity  follows. 


44  PROJECTIVITIES  OF  PRIME  FORMS 

Theorem.  Fundamental  Theorem  of  Prime  Forms 

40.  Betiveen  two  one-dimensional  prime  forms  there  exists 
one  and  only  one  projectivity  in  which  three  elements  of  one 
form,  in  a  specified  order,  correspond  to  three  elements  of  the 
other  form,  also  in  a  specified  order. 


Proof.    Let  us  first  consider  three  special  cases. 

1.  Suppose  that  the  two  prime  forms  are  a  fiat  pencil  and 
the  range  obtained  hy  cutting  the  pencil  by  a  line. 

Let  the  lines  of  the  flat  pencil  be  a,  6,  <?,  •  • .,  Z,  .  •  •  and 
let  these  lines  be  cut  by  a  line  p  in  the  points  A,B,C,''', 
L,  '  '  '.  Then  these  prime  forms  are  perspective  in  such  a 
way  that  a.  A',  b,  B;  c,  C;  '  •  •;  I,  L;  '  '  •  are  pairs  of  cor- 
responding elements. 

Assume  that,  if  possible,  a  second  projectivity  exists  in 
which  the  first  three  of  these  pairs  of  elements  correspond, 
but  in  which  I  corresponds  to  3f  and  not  to  L. 

Then,  from  the  perspectivity, 

(abcl)  =  (ABCL); 
and,  from  the  second  projectivity, 

(abcl)  =  (ABCM}. 
Then  (ABCL^^^ABCM), 

which  is  impossible  unless  L  =  M. 

Hence  the  second  projectivity  cannot  exist,  and  the  only 
projectivity  existing  between  the  forms  is  the  perspectivity. 


FUNDAMENTAL  THEOREM 


45 


2.  Suppose  that  the  two  prime  forms  are  an  axial  pencil 
and  the  range  obtained  hy  cutting  the  pencil  hy  a  line. 


Fig.  1 


Fig.  2 


Let  the  planes  of  the  axial  pencil  be  a,  /8,  7,    •  •  • ,  X,  •  •  • 
(Fig.  1),  and  let p  cut  the  planes  in  A,  B,  C,  •  •  -,  L,  •  •  -. 

A  proof  similar  to  that  on  page  44  should  be  given  by  the  student. 

3.  Suppose  that  the  two  prime  forms  are  ranges. 

Let  A^,  B^,  Cj  (Fig.  2)  be  three  points  of  the  first  range, 
and  let  them  correspond  respectively  to  the  points  A^,  B^,  C^ 
of  the  second.  A  projectivity  exists  between  these  sets  of 
three  points,  and  this  projectivity  can  be  extended  to  include 
the  whole  of  both  ranges.    Let  ij  correspond  to  L^. 

If  possible  let  there  be  a  second  projectivity  connecting 
the  ranges,  in  which  ij  corresponds  to  M^,  a  point  other 
than  Z-g.   Then,  from  the  two  projectivities, 

(A,B,C,L,')  =  (A^B^C,^L^') 

and  (^1^1  qZi)  =  (A^B^C^M^-). 

Therefore        (A^B^C^L^')  =  (A^B^C^M^), 
which  is  impossible  unless  L^  =  M^. 

Hence  in  this  case  two  projectivities  cannot  exist. 


46  PROJECTIVITIES  OF  PRIME  FORMS 

4.    Consider  now  the  general  case. 

Let  the  two  forms  be  /,  /',  and  let  the  three  pairs  of 
corresponding  elements  be  1,  1' ;  2,  2' ;  3,  3'. 

/ — r'^""c"'^D"^ 

r 

A'  B'   C   D'    E'. 


If  /  is  a  range,  let  the  elements  1,  2,  3,  4,  5,  .  •  •  be  A, 

B,  C,  D,  E,  • '  •;  but  if  /  is  not  a  range,  let  the  elements 
1,  2,  3,  4,  5,  .  •  •  be  cut  by  a  line  in  the  points  A,  B,  C, 
D,  U,  '  • '.  Similarly,  if  /'  is  a  range,  let  the  elements  1', 
2',  3',  4',  5',  .  .  .  be  A',  B',  C\  D',  E',...;  but  if/'  is  not  a 
range,  let  the  elements  1',  2',  3',  4',  5',  •  •  .  be  cut  by  a  line 
in  the  points  A',  B',  C\  D\E'  .... 

Between  the  ranges  ABCD  •  •  •  and  A'B'C'D'  .  .  .  there  is 
just  one  projectivity  in  which  4,  A'-,  B,  B' ;  C,  C  are  cor- 
responding elements.    Let  D,  D'  be  corresponding  elements. 

Then         form  1234  ...- range  ^J5Ci)  ••  • 

-  range  A'B'C'D'  ...  -  form  1'2'3'4',  .... 
Hence  form  1234  ...  -  form  1'2'3'4'  •  .  .. 

Suppose  that,  if  possible,  between  /  and  /'  there  is  a 
second  projectivity  in  which  1,  1';  2,  2';  3,  3'  are  pairs  of 
corresponding  elements  and  4,  5'  are  corresponding  elements. 

Then         range  ABCD...-  form  1234  .. ., 

and  form  1'2'3'5'  ...  -  range  A'B'C'E'  • . .; 

whence  range  ABCD  •  •  •  -r-  range  A'B'C'E'  .... 

Accordingly  there  would  be  a  second  projectivity  between 
the  ranges  ABC ...  and  A'B'C  .  .  .,  in  which  A,  A';  By  B'; 

C,  C  are  pairs  of  corresponding  points.  This,  however,  is 
impossible.    Hence  the  theorem  is  true  in  all  cases. 


fundame:ntal  theorem  47 

41.  Corollary.  There  is  one  projectivity  and  only  one 
vrojectivity  between  one-dimensional  forms  on  the  same  base 
which  makes  three  distinct  elements  of  a  one-dimensional  form 
correspond  each  to  itself.  This  projectivity  makes  every  element 
of  the  form  correspond  to  itself. 

Exercise  13.    Projectivities  of  Triads 

1.  li  A,  B^,  B^,  Cj,  Cj  ^i'6  fiv6  distinct  points  on  a  line,  find 
a  set  of  projections  and  sections,  minimum  in  number,  which 
connects  the  triads  AB^C^  and  ^52^2- 

2.  Examine  Ex.  1  for  the  case  in  which  B^  coincides  with  B^. 

3.  li  A,  B,  C  are  three  points  on  a  line,  find  the  set  of 
projections  and  sections,  minimum  in  number,  which  connects 
these  points  with  themselves  in  any  selected  order. 

Consider  Ex.  3  for  each  of  the  six  possible  orders  of  the  points. 

4.  If  A^,  A^,  B^,  -Bjj,  Cj,  Cj  are  six  distinct  collinear  points, 
find  a  set  of  projections  and  sections,  minimum  in  number, 
which  connects  the  triads  A^B^C^  and  A^^C^. 

5.  If  .4,  B^,  B^,  Cj,  Cg  are  five  points,  no  four  of  which  are 
coplanar,  find  a  set  of  projections  and  sections,  minimum  in 
number,  which  connects  the  triads  ABf!^  and  AB^C^. 

This  is  a  special  case  of  a  more  general  problem. 

6.  In  how  many  ways  can  a  projectivity  between  the  triads 
specified  in  Ex.  5  be  established  ? 

7.  If  A^,  A^,  5j,  B^,  Cj,  Cj  are  six  points  in  space,  no  four 
of  which  are  coplanar  and  no  three  of  which  are  collinear, 
the  triads  A^B^C^  and  Afi^C^  are  projective.  Specify  a  set  of 
projections  and  sections  which  constitutes  such  a  projectivity. 

8.  Investigate  the  possibility  of  establishing  a  projectivity 
between  the  triads  specified  in  Ex.  5  such  that  any  fourth 
point  Z>j  in  the  plane  AB^C^  shall  correspond  to  any  fourth 
point  Dj  in  the  plane  AB,^C^. 


48  PROJECTIVITIES  OF  PRIME  FORMS 

Theorem.  Metric  condition  for  a  projectivity 

42.  If  between  the  elements  of  two  one-dimensional  prime 
forms  there  exists  a  one-to-one  correspondence  such  that  the 
anharmonic  ratio  of  every  set  of  four  elements  of  one  prime 
form  is  equal  to  that  of  the  set  of  four  corresponding  elements 
of  the  other  prime  fonm^  the  correspondence  is  a  projectivity. 


Ai    B,     C,    D, 


Az     Bz  Cg     D2 

Proof.  It  is  sufficient  to  consider  the  case  of  two  ranges 
because  if  either  of  the  given  prime  forms  is  not  a  range, 
it  is  possible  by  operations  of  projection  and  section  to 
obtain  from  it  a  range  which  is  projective  with  it. 

From  §  39  it  follows  that  a  projectivity  exists  between 
three  points  Ay,  J5j,  C^  of  the  first  range  and  the  three 
corresponding  points  A^,  B^,  C^  of  the  second  range.  Then 
if  a  fourth  pair  of  corresponding  points  B^,  D^  on  the  ranges 
are  found,  it  follows  by  hypothesis  that 

(:a,b^c^d^)  =  Ca^b^c^d^'). 

Also  let  range  A-^Bj^C^D^^  —  range  A,^B^C^D'^. 

Then  (^2^2<^2^2)  =  (^1^1  CiA)'  §  26 

and  so  (A^B^C^d;,)  =  (A^B^C^D^). 

Accordingly  Z)^  coincides  with  D^ ;  and  therefore 

range  ^2-^2 ^2^2  a  ^^^^g^  A^i^i^r 

Thus,  any  fourth  point  -Z>j  of  the  first  range  has  the  same 
corresponding  point  D^  in  the  given  one-to-one  correspond- 
ence as  it  has  in  the  projectivity  between  the  ranges  which  is 
determined  by  the  triads  of  points  A-^B^C^^  and  ^3^2 ^2* 
Hence  the  correspondence  is  this  projectivity. 


METRIC  COXDITIOX  FOIl  A  PROJECTIVITY     49 

If  it  is  given  that  two  one-dimensional  prime  forms  are 
projective,  it  should  be  noted  that  it  is  necessary  only  that 
tliree  elements  of  one  of  them  and  the  corresponding  ele- 
ments of  the  other  be  specified  in  order  that  by  the  con- 
struction of  §  39  the  whole  projectivity  may  be  established ; 
and  this  construction  furnishes  a  standard  method  of 
establishing  a  projectivity. 

Associated  with  the  general  results  of  §§  39-42,  several  which  are 
special  in  their  nature  and  application  are  considered  in  §§  43-46. 

Exercise  14.    Projectivity  of  Prime  Forms 

1.  Given  three  points  A^,  B^,  C^  of  a  range  and  the  corre- 
sponding points  A^,  B^,  C^  of  a  second  range  projective  with 
the  first,  the  bases  being  different,  construct  the  point  D^  of 
the  second  range  corresponding  to  a  fourth  point  D^  of  the  first. 

2.  Consider  Ex.  1  when  the  point  D^  is  at  infinity. 

3.  Consider  Ex.  1  when  the  ranges  are  coplanar  and  D^ 
is  their  intersection. 

4.  Obtain  simplified  constructions  for  Exs.  1  and  2  when 
A^  and  A^  coincide  at  the  intersection  of  the  ranges. 

5.  Consider  Ex.  1  when  the  bases  are  coincident. 

6.  Consider  Ex.  5  when  A^,  B^;  A^,  B^;  C\,  D^  are  all  pairs 
of  coincident  points. 

7.  Prove  the  theorem  suggested  by  the  result  of  Ex.  6. 

8.  If  two  ranges  are  projective,  to  every  harmonic  range  in 
one  of  them  there  corresponds  a  harmonic  range  in  the  other. 

9.  Assuming  that  if  four  points  A,  B,  C,  D  are  properly 
divided  into  pairs  a  common  pair  of  harmonic  conjugates  with 
respect  to  these  pairs  exists,  prove  the  converse  of  Ex.  8. 

10.  Investigate  the  question  of  dividing  a  set  of  four  col- 
linear  points  into  pairs  so  that  a  common  pair  of  harmonic 
conjugates  may  exist. 


50 


PROJECTIVITIES  OF  PRIME  FORMS 


Theorem.  Projective  Ranges  in  Perspective 

43.   Tivo  projective  ranges  whose  bases  are  not  coplanar 
are  perspective, 

Co 


Proof.  Let  A^B^C^-  >  >  L^'  -  •  on  the  base  p^  and  ^2^2  ^2 
•  •  •  ij  •  •  •  0^^  the  base  p^  be  projective  ranges  in  which 
Ay,  A^ ;  5|,  B^\  Cj,  Cg ;  •  •  •  are  corresponding  elements, 
their  bases  not  being  coplanar. 

On  the  line  A.^A^  take  a  point  A^  and  let  C  be  the  point 
in  which  the  line  C^C^  intersects  the  plane  determined  by 
the  point  A  and  the  line  B^B^.  Let  the  line  AC  intersect 
the  line  B^B^  in  the  point  B. 

Denote  by  or,  y8,  7,  .  •  •,  \,  •  •  •  the  planes  determined  by  the 
line  ^J?Cand  the  points  Jp  5j,  Cj,  •  •  •,  ij,  •  •  •  respectively. 

The  axial  pencil  ay87  •  •  •  X  •  •  •  is  perspective  with  the 
range  A^B^C^ . . .  i^ . . .  and  cuts  the  line  jt?2  i^i  a  range  pro- 
jective with  the  range  A^B^^  Cj  •  •  •  X^  • .  • . 

Moreover,  in  the  projectivity  thus  established  the  points 
A^y  B^  Cg  correspond  to  the  points  ^j,  Bj,  C^  respectively. 
This  projectivity  is  therefore  the  same  as  the  one  that  was 
originally  assumed  to  exist  between  the  ranges  (§  40). 

Hence  from  the  range  A^B^C^  •  •  •  on  the  base  p^  it  is 
possible  to  pass  to  the  range  A^B^C^  •  • .  on  the  base  p^ 
by  one  operation  of  projection  from  the  axis  ABC  and 
one  operation  of  section  by  the  line  p^  Hence  tlie  pro- 
jectivity existing  between  the  ranges  is  a  perspectivity. 


PROJECTIVE  RANGES 
Theorem.  Condition  for  Perspective  Ranges 


51 


44.  Two  projective  ranges  on  different  bases  in  a  plane 
IT  are  perspective  if  and  only  if  the  point  common  to  them  is 
self-corresponding  in  the  projectivity. 


IT 

xi 

:?'' 

By 

'ca 

(■ 

i 

/ 

\ 

< 

\ 

Fig.  1 


Fig.  2 


Proof.  First,  let  the  ranges  A^B^C^  •  •  •  (Fig.  1)  on  the 
base  jOj  and  .42^2^2  •  •  •  on  the  base  p^  be  projective  in  such 
a  way  that  X,  the  intersection  of  the  two  ranges,  is  self- 
corresponding  in  the  projectivity. 

Let  the  lines  A^A^  and  B^B,^  intersect  at  P,  and  draw  PX. 

The  projectivity  is  completely  determined  by  the  corre- 
spondence of  Ay,  By  X  with  A^,  B^,  X,  but  this  correspond- 
ence is  established  by  the  projection  of  A^  B^  X  from  the 
center  P  and  the  section  of  the  resulting  flat  pencil  by 
the  line  p^.    Hence  the  ranges  are  perspective. 

Next,  let  the  ranges  be  perspective  (Fig.  2). 

Let  the  intersection  of  p^  and  p^,  regarded  as  a  point  of 
the  first  range,  be  denoted  by  X^  Since  its  correspond- 
ing point  would  be  found  by  cutting  the  projector  of  X^ 
from  a  center  P  in  the  plane  tt,  or  from  an  axis  p  not  in 
this  plane,  as  the  case  may  be,  by  the  line  p^,  it  is  the 
intersection  Xy    That  is,  X^  is  self-corresponding. 


62 


PROJECTIVITIES  OF  PRIME  FORMS 


Theorem.  Ranges  Perspective  with  a  Third  range 

45.  Two  projective  (but  not  perspective^  rallies  on  different 
bases  in  a  plane  are  both  perspective  with  the  same  range 
on  any  line  that  does  not  pass  through  the  intersection  of  the 
bases  of  the  given  ranges. 


Proof.  The  construction  used  in  Case  1  of  §  39  estab- 
lishes the  existence  of  a  range  which  is  perspective  with 
each  of  the  two  given  coplanar  ranges.  It  is  now  necesgary 
to  show  that  the  points  P^  and  P^  can  be  so  selected  that 
the  line  -45  C  shall  be  any  line  which  does  not  pass  through 
the  intersection  of  p^  and  p^. 

Let  p  be  any  line  not  concurrent  with  p^  and  p^.  Let 
the  intersection  of  p^  and  p  be  i>j,  and  that  of  p^  and  p 
be  E^.  Suppose  that  the  point  D^  corresponding  to  Z)j,  and 
the  point  E^  corresponding  to  E^,  have  been  found  by  the 
method  of  Case  1  of  §  39,  or  by  any  other  suitable  method. 
The  problem  now  is  to  choose  points  P^  and  P^  such  that 
the  line  ABC  shall  coincide  with  the  line  p  or  D^E^ 

and   let  this   line   meet  A^A^  in  j^.    Draw 
and  let  this  line  meet  A.  A.,  in  B.    Join  P  to  B., 


Draw  Dj^j, 


^i^v 


■  ,  and  let  these  lines  meet  DiE^  in  B,  C, 


PERSPECTIVE  RANGES  53 

Then     range  ABCD^E^  .  •  .  =  range  A^B^C^D^E^  .... 
Hence   range  ABCD^E^  ' ' '  a  ^^g^  ^^2^2^2^2  *  *  '• 

But  the  triad  AD^E^  is  perspective  with  the  triad  A^D^E^y 
and  these  triads  determine  completely  the  nature  of  the 
projectivity  between  the  range  ABCD^E^  • .  .  and  the  range 

^2^2^2^2^2  '  •  '• 

Therefore  the  two  ranges  A^B^C^D^E^ ...  on  jOj  and 
A^B^C^D^E^ . . .  on  jt?2  ^-^e  perspective  with  a  range  on 
the  given  line  p. 

Exercise  15.    Projective  and  Perspective  Forms 

1.  State  and  prove  the  dual  of  §  43. 

State  and  prove  the  duals  of  each  of  the  following : 

2.  §  44,  in  the  plane.  5.  §  45,  in  the  plane. 

3.  §  44,  in  space.  6.  §  45,  in  space. 

4.  Ex.  3,  in  the  bundle.  7.  Ex.  6,  in  the  bundle. 

Solve  the  duals  of  each  of  the  following  : 

8.  Ex.  1,  page  49,  in  space. 

9.  Ex.  3,  page  49,  in  the  plane. 

10.  Ex.  3,  page  49,  in  space. 

11.  Ex.  1,  page  49,  in  the  plane,  considering  the  case  in 
which  the  ranges  are  coplanar. 

12.  If  three  ranges  in  a  plane  have  concurrent  bases  and  if 
two  of  the  ranges  are  perspective  with  the  third,  these  two 
ranges  are  perspective  with  each  other. 

13.  Three  fixed  lines  Pi,p^,p^  radiate  from  A^,  one  of  three 
fixed  collinear  points  A^,  A^,  A^.  A  point  P^  moves  on^^,  and 
the  lines  A^P^,  A^P^  cut  ^j^j  2\  respectively  in  P^,  P^.  Show  that 
P^P^  has  a  fixed  point. 


54  PROJECTIVITIES  OF  PRIME  FORMS 

46.  Special  Case.  In  the  construction  described  in  §  39 
the  only  hmitation  upon  the  position  of  the  points  ^  and  7^ 
was  that  ^  should  not  be  at  ^^  and  that  J^  should  not  be  at 
A^'  The  necessity  for  this  limitation  is  due  to  the  complete 
failure  of  the  construction  which  would  result  from  the 
coincidence  of  the  lines  ij^i,  ^^i,  ^C^  We  shall  now 
consider  a  special  case  which  is  noteworthy  because  it 
puts  in  evidence  a  line  that  has  important  relations  to  all 
pairs  of  corresponding  points  of  the  ranges  and  furnishes 
a  basis  for  several  simple  constructions. 


Pi 


Let  ^  be  taken  at  A^,  and  ^  at  Ay  Then  the  line  p  is 
determined  by  the  intersection  of  I\B^(^A^B^^  and  I^B^ 
(^A^B^  and  the  intersection  of  I^C^^^A^C^)  and  ^C^i^A^^C^}' 
It  contains  likewise  the  intersections  of  A^D^  ^1^2  >  ^2-^1' 
A^E^ ;  and  so  on.  It  can  now  be  shown  that  this  line  p 
can  be  located  independently  of  the  placing  of  ^  and  ^ 
on  the  particular  line  A^A^. 

If  the  ranges  A^B^C^  '  •  •  and  A^B^C^  •  •  •  are  projective 
but  not  perspective,  the  point  of  intersection  of  p^  and  p^ 
will  not  be  self-corresponding.  Regarding  this  point  as 
belonging  to  the  range  A^B^C^  •  •  -,  call  it  Xj.  In  the  other 
range  there  will  correspond  to  X^  a  point  Xj.  Regarding 
the  intersection  as  a  point  of  range  A^B^C^  •  •  .,  call  it  Y^. 
To  it  will  correspond  a  point  Y^  of  the  first  range.  Then 
^jXj  and  ^2^1  intersect  on  the  line  p.  But  their  inter- 
section is  Xj. 


PERSPECTIVE  RANGES  55- 

Similarly,  Y^,  the  intersection  of  A^Y^  and  A^Y^,  is  on  the 
line  jt).  Accordingly  the  line  p  is  the  line  determined  by  the 
points  of  the  first  and  second  ranges  which  correspond  to 
the  point  of  intersection  of  the  ranges  regarded  as  a  pomt 
of  the  second  and  first  ranges  respectively.  It  follows  that 
the  line  p  is  determined  by  the  projectivity  between  the 
ranges  and  bears  the  same  relation  to  any  two  correspond- 
mg  points  of  the  ranges  that  it  does  to  A^,  A^.  Accordingly 
not  only  do  A^B^,  A^B^;  A-^C^^  A^C^;  A^D^,  A^D^\  ••• 
intersect  on  p,  but  so  do  B^  C^i  B^  C^ ;  B^D^,  ^2^1 5  *  *  *  5 
CjDg,  C^D^;  C^E^,  C^E^',  .••;  and  so  on. 

If  the  ranges  A^B^C^  —  and  A^B^C^  •  •  •  are  perspective, 
their  common  point  X  is  self -corresponding.  Let  p  be  the 
line  joining  X  to  the  intersection  of  A^B^  and  A^B^ 

We  then  see  that  the  flat  pencils  A^(^A^B^C^  •  •  •  X*  •  •) 
and  A^^A^Bj^C^  •  •  •  A'.  •  •)  are  projective  and  have  a  com- 
mon line  Aj^A^.  Hence  these  pencils  are  perspective,  and 
the  axis  of  perspective  is  p.  Hence,  as  in  the  other  case, 
Aj^B^,  A^B^ ;  A-^ Cg,  A^C^ ;  A^D^^  A^D^ ;  >  ";  B^C^,  B^C^;  • .  •; 
(7jZ>2,  C^D^  all  intersect  on  p. 

Exercise  16.    Perspective  Forms 

1.  If  a  simple  reentrant  hexagon  has  its  first,  third,  and 
fifth  vertices  on  one  straight  line  of  a  plane  tt  and  has  its 
second,  fourth,  and  sixth  vertices  on  a 
second  straight  line  of  that  plane,  the 
intersections  of  the  first  and  fourth, 
the  second  and  fifth,  and  the  third  and 
sixth  sides  are  collinear. 

2.  State  and  prove  the  proposition  dual  in  a  plane  to  Ex.  1. 

3.  State  and  prove  the  proposition  dual  in  space  to  Ex.  1. 

4.  State  and  prove  the  proposition  dual  in  the  bundle  to  Ex.  3 


56  PROJECTIVITIES  OF  PRIME  FORMS 

Problem.  Line  to  an  Inaccessible  Point 

47.  To  draw  a  line  which  shall  pass  through  the  ifiacces- 
sible  intersection  of  two  given  straight  lines  and  also  through 
a  given  point  0  of  their  plane. 

P 


Solution.  To  draw  a  line  which  shall  satisfy  the  first 
condition,  let  the  given  lines  be  p^  and  p^.  Select  a  point 
P  in  the  plane  of  the  lines  p^  and  jp^,  but  not  on  either 
line,  and  through  P  pass  three  lines  cutting  p^  in  ^j,  B^,  Cj 

and  ^2  ^  ^2'  -^2'  ^2- 

Since  the  ranges  Ylji5jCj  and  A^B^C^  are  perspective, 
the  pairs  of  lines  A^B^,  A^B^;  A^C^,  ^2^v  ^\^i->  ^2^1  ii^ter- 
sect  on  a  line  p  which  passes  through  the  intersection  of 
the  lines  p^  and  p^  (§  46). 

Now  to  satisfy  the  second  condition  we  must  so  choose 
the  point  P  that  the  line  p  will  pass  through  0. 

Through  0  draw  two  lines  and  let  them  intersect  p^  in 
A^  and  B^  and  intersect  p^  in  A^  and  B^.  Take  P  to  be 
the  intersection  of  A^A^  and  B^B^^,.  Through  P  draw  an- 
other line  cutting  p^  in  C^  and  p^  in  C^.  Then  the  line  p 
which  is  determined  by  the  intersections  of  two  of  the  three 
pairs  of  lines  A^B^,  A^B^;  A^C^^  A^C^;  B^C„,  B^C^  satisfies 
the  two  given  conditions. 


PROBLEMS  OF  CONSTRUCTION  67 

Problem.  Line  Parallel  to  a  Given  Liwe 

48.  Given  two  parallel  lines  and  a  point  in  their  plane, 
to  draw  through  the  point  a  line  parallel  to  the  given  lines. 

The  solution  is  left  for  the  student,  who  should  write  out  the 
proof  after  the  method  used  in  §  47. 

The  case  in  §  49  is  somewhat  different. 

PROBLEM.  Line  Parallel  to  a  Given  Line 

49.  Gfiven  in  a  plane  a  point  0,  a  line  p^  not  passing 
through  0,  and  a  parallelogram  none  of  whose  sides  is  known 
to  be  parallel  to  p^,  to  draw  through  0  a  line  parallel  to  p^ 


Pi  Ai 


Solution.  This  problem  may  be  reduced  to  the  preced- 
ing one  as  follows: 

Let  the  adjacent  sides  a^  and  Jj  of  the  parallelogram 
meet  p^  in  A^  and  B^^  respectively.  Let  C  be  any  point  on 
the  diagonal  I\P^,  and  let  the  lines  CAj^  and  CB^  meet  a^ 
and  ^2  ill  ^2  ^^^  -^2  respectively. 

Then  the  triangles  J^A^B^  and  F^A^B^  are  homologic,  and 
since  the  intersections  of  -Pi^i,  ^^2-^2  ^^^  ^^  ^i^v  -^2^2  ^^^ 
at  infinity,  the  axis  of  homology  is  at  infinity. 

Hence  A^B^  is  parallel  to  p^. 

We  now  have  two  parallel  lines  A^B^  and  p^,  and  by 
means  of  §  48  we  can  draw  through  the  given  point  0  a 
line  parallel  to  the  given  hue  p^. 


58  PROJECTIVITIES  OF  PRIME  FORMS 

Theorem.  Similar  Ranges 

50.  If  in  ttoo  projective  ranges  the  points  at  infinity  are 
corresponding  points^  the  ratios  of  all  pairs  of  corresponding 
segments  are  the  same,  and  coiiversely. 

Proof.  Let  the  points  at  infinity  of  two  ranges  be  J^ 
and  J^,  and  suppose  that  the  points  t/j,  A^,  J?^,  Cj,  •  •  •  of 
tlie  first  range  correspond  respectively  to  the  points  J^, 
A^i  B^,  C2,  • ' '  of  the  second  range. 

Then  (A^B^C^J^)  =  (A^iVi). 

or  A^C^-.B^C^^A.C.iB.C,.  §22 

Hence  ^g  Cg  :  ^  1 C^  =  ^2  ^"2  ^  ^1  <^i- 

Similarly,  it  can  be  shown  that 

A^B^:  A^Bi  =  A^C^:  A,C,  =  A^D^:  A,D,=  •  .  . 

=  B^C^:  B^C\  =  B^I)^:  B,B,=  .  .  .  =  r, 

where  r  is  independent  of  the  pair  of  corresponding  seg- 
ments involved. 

Conversely,  suppose  that  all  the  pairs  of  corresponding 
segments  have  the  same  ratio. 

Let  A^,  A^;  B^  B^;  C^,  C^\  Jj,  K^  be  pairs  of  correspond- 
ing points,  t/j  being  the  point  at  infinity  of  the  first  range. 

Then  (^1^1  Vi)  =  {A^B^C^K^), 


or 


-^1^1 ^2^2 

.  AK^ 

-^1^1      -^2^2 

B^K^ 

But 

A,C, 

:  ^2^2  =^1^1  ' 

B^C^. 

Therefore 

A^iq : 

B^K^=\, 

and  hence  K<^  must  be  at  infinity  and  should  be  called  J^. 
Accordingly  J^  and  J^,  the  points  at  infinity  of  the  two 
ranges,  are  corresponding  points. 


SIMILAR  AND  CONGRUENT  FORMS  59 

51.  Similar  Ranges.  Two  projective  ranges  whose  points 
at  infinity  are  corresponding  points  are  said  to  be  similar. 

An  example  of  similar  ranges  is  furnished  by  sections  of  a  flat 
pencil  by  parallel  lines,  or  by  sections  of  a  flat  pencil  of  parallel  rays 
by  any  two  lines. 

If  similar  ranges  are  on  the  same  base,  the  point  at 
infinity  is  a  self-corresponding  point.  If  the  ratio  of  corre- 
sponding segments  (§  50)  is  1:1,  there  is  no  other  such 
point,  but  in  any  other  case  a  second  point  exists. 

52.  Congruent  Ranges.  Similar  ranges  in  which  the  ratio 
of  corresponding  segments  is  unity  are  said  to  be  congruent. 

If  two  similar  ranges  have  parallel  bases,  their  common  point  at 
infinity  is  self-corresponding  and  the  ranges  are  perspective. 

53.  Similar  and  Congruent  Pencils.  The  terms  similar 
and  congruent  are  applied  to  certain  special  cases  of  flat 
pencils  and  axial  pencils,  the  mere  mention  of  this  fact 
being  sufficient  for  the  present  treatment. 

Two  projective  flat  pencils  whose  bases  are  at  infinity 
are  said  to  be  similar  if  linear  sections  of  these  pencils 
are  similar  ranges. 

In  this  and  in  similar  cases  the  student  should  draw  the  figure. 

Two  projective  flat  pencils  whose  bases  are  in  the  finite 
part  of  the  plane  are  said  to  be  equal  or  congruent  if  every 
pair  of  lines  of  one  pencil  contains  an  angle  equal  to  the 
angle  contained  by  the  corresponding  pair  of  lines  of  the 
other  pencil. 

Similar  and  congruent  axial  pencils  are  defined  in  the  same  way. 

Flat  pencils  and  axial  pencils  are  said  to  be  proper  or 
improper  according  as  their  bases  are  -or  are  not  in  the 
finite  part  of  space. 


60  PROJECTIVITIES  OF  PRIME  FORMS 

Exercise  17.    Projectivities 

1.  There  exist  infinitely  many  sets  of  projections  and  sec- 
tions which  connect  a  prime  form  with  itself. 

2.  There  exist  infinitely  many  sets  of  projections  and  sec- 
tions which  connect  two  projective  prime  forms. 

3.  A  plane  quadrangle  is  projective  with  a  properly  chosen 
parallelogram. 

4.  A  plane  quadrangle  is  projective  with  a  properly  chosen 
square. 

5.  A  plane  quadrangle  is  projective  with  every  square. 

6.  Every  two  plane  quadrangles  are  projective.  Specify 
a  set  of  operations  that  connects  the  quadrangles. 

7.  A  triangle  and  any  point  in  its  plane,  but  not  in  its 
perimeter,  are  projective  with  any  equilateral  triangle  and 
its  center. 

8.  Solve  the  duals  of  Ex.  6. 

9.  Two  ranges  on  the  same  base  are  projective  if  they  are 
so  related  that  every  pair  of  corresponding  points  is  a  harmonic 
conjugate  with  respect  to  a  given  pair  of  points  on  the  base. 

Apply  Ex.  12  on  page  30. 

10.  If  Jj  and  K^  are  the  points  at  infinity  of  two  projective 
ranges  A^B^C^  •  •  •  and  AJi^C^  -  •  •)  J^  and  K^  being  the  points 
corresponding  to  them,  then  A^^K^  •  A^J^  =  B^K^  ■  B^J^  =  •  •  ■. 

11.  If  the  ranges  in  Ex.  10  are  on  the  same  base,  and  if  0^ 
is  the  midpoint  of  ./^A'^  and  if  O^  is  the  point  corresponding  to  0^, 
then  0/, .  O^A^  -  0/,  •  O/,  -  O^K^  •  O^A^  +  O^K^  •  Ofi^  =  0. 

12.  If  A^  and  A^  in  Ex.  11  coincide  with  A,  it  follows  that 

13.  If  P^  is  a  fixed  point  on  the  side  PJ^^  of  a  given  triangle 
PjPjPg,  and  if  a  moving  line  cuts  P^P^  in  P^,  P^P^  in  Q,  and 
PJ'^  in  R,  P^Q  and  P^R  meeting  in  P,  then  P^P  cuts  P^P^  in 
a  fixed  point. 


PROJECTIVITIES  61 

14.  Generalize  Ex.  12,  page  53. 

15.  Given  A^B^C^ . .  •,  A^^^  •  •  -,  and  A^B^C^  •  •  •,  three  pro- 
jective but  not  perspective  ranges  whose  bases  p^,  p^,  p^  are 
coplanar  and  concurrent,  prove  that,  if  any  triad  of  corre- 
sponding points  is  collinear,  there  exists  a  range  ABC  •  •  • 
which  is  perspective  with  each  of  the  three  given  ranges. 

16.  If  Pj,  P^,  Pg  in  Ex.  15  are  the  centers  from  which  the 
three  ranges  on  p^,  p^,  p^  respectively  are  projected  into  the 
range  ABC  ■  •  >,  each  of  the  sides  of  the  triangle  P^P^P^  cuts 
a  pair  of  the  given  ranges  in  points  which  correspond  in  one 
of  the  projectivities. 

17.  If  in  Ex.  16  P  moves  along  the  base  p  of  the  range 
ABC  •  ■  •,  the  lines  P^P,  P^P  and  P^P  trace  on  any  fourth  line 
of  the  plane  three  projective  ranges  which  have  one  point  that 
is  self-corresponding  for  all  three  projectivities. 

18.  If  Xj,  X^ ;  Fj,  F2 ;  A^,  A^axe.  pairs  of  corresponding  points 
of  two  coplanar  projective  ranges,  and  if  (§  46)  X^  and  Y^  coin- 
cide at  the  intersection  of  the  bases,  determine  the  position 
of  B^  which  corresponds  to  any  fourth  point  B^  of  the  first 
range,  basing  the  solution  on  §  46. 

19.  Solve  the  plane  dual  of  Ex.  18. 

20.  Solve  the  space  dual  of  Ex.  18. 

In  solving  the  remainder  of  the  problems  in  this  exercise,  use  only  the 
ungraduated  ruler.  These  exercises  are  chiefly  due  to  Steiner,  one  of  the 
founders  of  the  science  of  projective  geometry. 

21.  Given  a  segment  AB  extended  to  twice  its  length,  divide 
it  into  any  number  of  equal  parts. 

22.  In  an  indefinite  straight  line  given  a  segment  AB 
divided  at  C  in  the  ratio  of  two  given  whole  numbers,  draw 
through  any  given  point  a  line  parallel  to  the  given  line. 

23.  Given  two  parallel  lines  ^  and^',  and  given  on^  a  seg- 
ment AB,  from  any  given  point  on  p  lay  off  a  segment  of  p 
equal  to  any  given  multiple  of  AB. 

24.  Divide  AB  va.  Ex.  23  in  the  ratio  of  two  given  integers. 


62  PROJECTIVITIES  OF  PEIME  FORMS 

25.  Given  a  parallelogram,  divide  any  segment  in  its  plane 
in  the  ratio  of  two  given  integers. 

26.  In  a  plane,  given  three  parallel  lines  which  cut  a  fourth 
line  in  the  ratio  of  two  given  integers,  draw  through  a  given 
point  a  line  parallel  to  a  given  line. 

27.  If  two  parallel  line  segments  which  have  to  each  other 
the  ratio  of  two  given  integers  are  given,  draw  a  line  parallel 
to  a  given  third  line. 

28.  Given  two  parallel  lines  and  a  segment  divided  in  the 
ratio  of  two  given  integers,  draw  through  a  given  point  a  line 
parallel  to  a  given  fourth  line. 

29.  Given  two  nonparallel  segments,  each  divided  in  the 
ratio  of  two  given  integers,  draw  a  line  parallel  to  a  given  line. 

30.  Given  a  square,  a  point,  and  a  line,  all  in  one  plane, 
draw  through  the  point  a  line  perpendicular  to  the  given  line. 

31.  Given  a  square  and  a  right  angle,  both  in  the  same 
plane,  bisect  the  angle. 

32.  Given  a  square  and  an  angle,  both  in  the  same  plane, 
construct  any  multiple  of  the  angle. 

In  each  of  the  following  problems,  in  addition  to  the  data  mentioned, 
one  circle  fully  drawn  and  its  center  are  assumed  to  be  given. 

33.  Through  a  given  point  draw  aline  parallel  to  a  given 
line. 

34.  Through  a  given  point  draw  a  line  perpendicular  to  a 
given  line. 

35.  Through  a  given  point  draw  a  line  which  shall  make 
with  a  given  line  an  angle  equal  to  a  given  angle. 

36.  From  a  given  point  draw  a  line  parallel  and  equal  to 
a  given  line. 

37.  Determine  the  intersections,  if  any,  of  a  given  line  and 
a  circle  of  given  center  and  radius. 

38.  Determine  the  intersections,  if  any,  of  two  circles  of 
given  centers  and  given  radii. 


CHAPTER  YII 

SUPERPOSED  PROJECTIVE  FORMS 

54.  Superposed  Projective  Forms.  Hitherto  only  inci- 
dental reference  has  been  made  to  the  existence  of  pro- 
jective forms  on  the  same  base,  but  the  general  results 
obtained  apply  to  them  except  when  the  contrary  is  in- 
dicated. It  is  proposed  now  to  consider  some  special 
properties  of  one-dimensional  projective  prime  forms  on 
the  same  base.    Such  forms  are  called  superposed  forms. 


In  the  first  place,  the  existence  of  superposed  projective 
forms  is  established  if,  when  two  projective  ranges  are  on 
different  bases,  as  A^B^C^  •  •  •  on  the  base  p^  and  A^B^C^  •  •  • 
on  the  base  p^,  the  second  range  is  projected  on  the  base  p^ 
from  a  center  P  so  taken  as  not  to  be  on  the  line  A^A^. 
The  result  is  a  range  A[B[C[  •  •  •  on  the  base  p^  which  is  pro- 
jective but  not  identical  with  Jj^jCj  •  •  •,  since  A^  and  A\ 
are  not  coincident. 

In  the  second  place,  if  each  of  two  superposed  projective 
forms  is  operated  upon  by  section  or  projection,  the  result- 
ing forms  are  superposed  and  projective. 

63 


64  SUPERPOSED  PROJECTIVE  FORMS 

Theorem,  existence  of  superposed  Projective  Forms 

55.  There  exist  pairs  of  projective  prime  forms  which 
have  common  bases  and  which  have  not  all  their  corre- 
sponding elements  coincident. 

The  proof  of  this  theorem  is  evident  from  §  54. 

Theorem.    Invariance  under  Projection 

56.  If  from  two  superposed  projective  prime  forms  two 
other  superposed  prime  forms  are  obtained  by  projection  or 
by  section^  the  latter  forms  are  projective. 

The  proof  of  this  theorem  is  evident  from  §  54. 

57.  Self-corresponding,  or  Double,  Elements.  It  has  been 
shown  (§  39)  that  three  points  A^^  B^,  C^  of  a  Ime  are 
projective  with  any  three  points  of  the  same  line,  even 
though  some  or  all  of  the  two  sets  of  three  are  the  same ; 
and  it  follows  from  §  40  that  the  correspondence  of  the 
three  pairs  of  points  establishes  for  all  points  of  the  line  a 
projectivity  in  which  some  of  the  points  may  coincide  with 
their  corresponding  points.  It  is  evident  that  similar  con- 
siderations apply  to  flat  pencils  and  to  axial  pencils. 

Elements  of  superposed  projective  prime  forms  that 
coincide  with  those  to  which  they  correspond  are  called  self- 
corresponding  elements,  or  double  elements ;  and  the  deter- 
mination of  the  number  of  such  elements  is  a  problem 
of  importance.  Since  from  superposed  flat  pencils  or 
axial  pencils  we  may  by  section  obtain  superposed  ranges 
whose  self-corresponding  points  are  situated  on  the  self- 
corresponding  elements  of  the  pencils,  the  discussion  of  this 
question  is  substantially  the  same  for  all  one-dimensional 
prime  forms,  and  hence  will  be  limited  to  the  case  of 
superposed  ranges. 


SELF-CORRESPONDING  ELEMENTS  65 

Theorem.  Self-corresponding  Elements 

58.  The  number  of  self -corresponding  elements  of  two  dis- 
tinct superposed  projective  one-dimensional  prime  forms  is 
two,  one,  or  none,  and  all  three  of  these  cases  occur. 

Proof.    The  proof  consists  of  the  following  four  parts : 

1.  There  may  he  two  self -corresponding  elements. 

If  A^,  -Bj,  Cj,  C^  are  four  points  on  a  line  p,  the  triads 
Jji?jCj  and  A^B^C.^  determine  a  projectivity  between  dis- 
tinct ranges  on  p  with  A^  and  B^,  but  not  with  either  Cj 
or  Cg,  as  self-corresponding  points. 

2.  There  may  he  just  one  self-corresponding  element. 


On  a  line  p  take  four  points  A,  B^,  B^,  Cj,  and  through 
A  pass  two  lines  a  and  p'.  On  a  take  any  point  J^.  Let 
the  line  I^B^  cut  p'  in  B',  the  line  B'B^  cut  a  in  ^,  the  line 
I^Ci  cut  p'  in  C",  and  the  line  ijC  cut  p  in  C^. 

The  triads  AB^C^  and  AB^C^  determine  a  projectivity 
between  distinct  ranges  on  p  in  which  the  point  A  is  self- 
corresponding.  Moreover,  each  of  these  ranges  on  p  is 
perspective  with  the  range  AB'C'  •  •  •  on  j9'  and  hence,  if 
?7j  and  U^  are  corresponding  points  of  the  range  on  p^,  it 
follows  that  I{  U^  and  I^  U-^  must  intersect  on  p'.  This  would 
not  happen  if  fTj  and  U^  were  coincident,  except  at  A. 
Therefore  A  is  the  only  self -corresponding  point. 


6(>  SUPERPOSED  PROJECTIVE  FORMS 

3.  There  may  he  no  self-corresponding  element. 

Taking  a  range  A^B^C^  •  --  on  2i  base  p,  let  its  projector 
from  a  point  P,  not  in  the  base,  be  the  pencil  a^h^c^  •  ■. 
Tlu-ough  P  clraw  the  lines  a^,  b^,  c^,  -  - -,  making  any  fixed 
angle,  say  30°,  with  ftp  b^,  c^,  -  •  -  respectively,  and  let  these 
lines  meet  the  base  p  in  A^,  B^,  Cj,  •  •  •. 

Then      range  A^B^C^  '"a  P^^^il  a^b^e^^ .  • . 

-  pencil  a^b^c^  • .  • 

-  range  ^2^2  ^2  '  '  - 

Since  the  ranges  A^B^C^  -  •  •  and  ^2-^2^2  *  * '  ^^^^®  "^  ^^^' 
responding  element,  this  part  of  the  theorem  is  proved. 

4.  There  cannot  be  three  self-corresponding  elements. 

It  follows  from  §  41  that  if  three  elements  are  self- 
correspondmg,  all  elements  are  self-corresponding,  and  the 
forms  are  coincident.  Hence  if  the  forms  are  distinct,  there 
cannot  be  three  self-corresponding  elements. 

In  selecting  the  triads  of  elements  which  determine  the  projec- 
tivity  between  superposed  forms  we  may  include  self-corresponding 
elements.  Thus,  for  ranges  the  projectivity  is  determined  if  the  two 
self-corresponding  points  X,  Y  and  a  pair  of  corresponding  points 
Ay  A^  are  given,  for  then  the  triads  of  corresponding  points  are 
A'F^i  and  XYA^.  Also,  if  one  self-corresponding  point  X  and  two 
pairs  of  corresponding  points  A^  A^;  B^,  B^  are  given,  the  deter- 
mination is  complete.  Here  the  triads  are  XA^B^  &nAXA^B^.  In  each 
of  these  cases  simple  constructions  serve  to  determine  additional 
pairs  of  corresponding  points. 

In  this  connection  the  student  may  review  the  solutions  of  Exs.  1 
and  2,  page  47,  which  furnish  constructions  for  these  cases. 

59.  Classes  of  Projectivities.  The  projectivity  between 
superposed  forms  is  said  to  be  hyperbolic  when  there  are 
two  self-corresponding  elements,  parabolic  when  there  is 
one,  and  elliptic  when  there  is  none. 


HYPERBOLIC  PROJECTIVITY  67 

Theorem.  Anharmonic  Ratio 

60.  If  tivo  superposed  projective  ranges  have  two  self- 
corresponding  points,  the  anharmonic  ratio  of  these  two  points 
with  any  pair  of  corresponding  points  is  independent  of  the 
choice  of  the  latter. 

Proof.  Let  X,  Y  be  self-corresponding  points  of  two 
superposed  projective  ranges,  and  let  A-^,  A^;  B^,  B^  be 
any  two  pairs  of  corresponding  points. 

Then  {XYA^B{)  =  (^XYA^B^^. 

mi       p              XAh    XB,      XAn    XBq 
Therefore : = : , 


and  hence 


YA^     YB^       YA^    YB^ 
XA^  ^  XA^  _  XB^    XB^ 


YA^    YA^      YB^    YB^ 
Therefore         (XYA^A^}  =  (XYB^B^). 


Exercise  18.    Superposed  Ranges 

Given  a  range  XYA^B^  •  •  • ,  find  two  points  A^,  B^  of  a 
range  on  the  same  base,  projective  with  the  given  range,  such 
that  X,  Y  shall  he  self-corresponding  points  and  the  ratio 
(^XYA^A^  shall  he  as  follows: 

1.  4.  2.  -4.  3.  -1.  4.  L 

6.  Consider  Exs.  1  and  3  when  Fis  the  point  at  infinity. 

In  Exs.  1-5  observe  the  situation  of  pairs  of  corresponding  points  with 
respect  to  the  self -corresponding  points. 

6.  Construct  two  superposed  ranges  in  which  the  point  at 
infinity  shall  be  the  only  self-corresponding  point. 

7.  Given  a  self-corresponding  point  and  two  pairs  of  corre- 
sponding points,  construct  the  other  self-corresponding  point 
of  two  superposed  projective  ranges. 


68  SUPERPOSED  PROJECTIVE  FORMS 

Theorem,  congruence  of  projective  Ranges 

61.  Two  superposed  projective  ranges  that  have  the  point  at 
infinity  as  their  only  self-corresponding  point  are  congruent. 

Proof.  Let  A^B^C^  •  •  •  and  A^B^C^  •  • .  be  two  projective 
ranges  on  the  same  base  jo,  the  only  self-corresponding 
point  of  the  ranges  being  the  point  J  at  infinity.  Let  X^ 
and  Aj  be  corresponding  points  not  at  infinity. 

Then         (^A^B^X^J^  =  {A^B^X^J^  ; 
whence  ^  =  ^.    .  §22 

Therefore  ^1^  =  ±&.-I=d^ -\  =  ^&, 

i?jAj      B^X^  B^X^  B^X^ 

A,B,_B,X, 


or 


^2^2        ^2^2 


Suppose  now  .that,  if  possible,  X^  is  taken  so  that 
B^X^:  B^X^  =  A^B^:  A^B^. 

Then  Mi  =  JJa^, 

B^X-^      B^X^ 

and  hence  X^  and  Xj  must  coincide. 

But  this  yields  a  self-corresponding  point  distinct  from 
t/,  which  is  contrary  to  the  hypothesis. 

Hence  the  choice  of  X^  in  the  finite  part  of  the  line,  as 
assumed  above,  must  be  impossible.  But  it  is  possible 
unless  A^B^ :  A^B^^l. 

A  B 
Hence  —^-^  =  1,  or  A^B^  =  A^B^ 

A^B^ 

It  follows  then  that  any  two  corresponding  segments 
are  equal  and  that  the  ranges  are  congruent. 
An  interesting  alternative  proof  is  given  in  §  62. 


CONGRUENCE  OF  PROJECTIVE  RANGES       69 

62.  Alternative  Proof.  Let  jo'  be  a  line  parallel  to  p, 
and  hence  intersecting/*  at  J.  Let  the  range  A^B^  —  J" be 
projected  from  any  center  P^  upon  the  base  p\  the  resulting 
range  being  A[B[  -  •  -  J. 

Because  the  latter  range  is 
perspective  with  A^B^  •  '  >  J, 
the  point  J  at  infinity  must 
be  self-corresponding  (§  44). 
Therefore  the  range  A^B^  ...  J" 
is  projective  with  the  range 
A[B'^  ...  J"  in  such  a  way  that  the  point  J  at  infinity  is 
self-corresponding,  and  hence  these  ranges  are  perspective. 

The  center  of  perspective  of  these  ranges  must  be  on 
A[A^.  Let  it  be  !{.  Then  B^  must  be  the  intersection  of 
B[P^  and  p.    Draw  P^P^. 

It  will  be  proved  that  I^  is  parallel  to  p  and  p'. 

If  P^P^  is  not  parallel  to  p  and  p',  let  it  meet  p  m  X^ 
and  p'  in  X[.    Then 

range  A^B^  .  .  .  Xj  •  •  •  J^  range  A[B[  .  •  •  X{  •  •  •  J" 

=  range  A^B^  ...  X^  ...  J". 

Hence  the  given  ranges  have  two  self-corresponding 
points  Xj  and  J,  which  is  contrary  to  the  hypothesis,  and 
so  I^I^  is  parallel  to  p. 

From  similar  triangles  it  follows  that 

A^B,^P^A,_P,A^_A^B^ 
A[B[      P,A[      P,A[      A'X 
and  hence  that  ^i  A  —  ^2^2* 

Similarly,  any  two  corresponding  segments  are  equal. 
Hence  the  ranges  are  congruent. 

Compare  the  above  figure  with  the  one  in  §  58. 


\ 


70  SUPERPOSED  PROJECTIVE  FORMS 

Theorem.  Angle  subtended  by  Corresponding  Points 

63.  Given  two  superposed  projective  ranges  having  no 
self-corresponding  point,  it  is  possible  to  find  a  point  at 
which  all  pairs  of  corresponding  points  in  the  range  subtend 
equal  angles. 

Proof.  Let  A^B^C^  •  •  •  and  ^2^2 ^2  •  •  •  be  projective 
ranges  on  a  base  jt),  and  let  Jj  and  K^  be  the  point  at 
infinity  of  p.  Let  K^  and  J^  be  the  points  of  the  two 
ranges  which  correspond  to  the  point  at  infinity. 

Bisect  -fiTjJj  in  Oj,  and  let  0^  of  the  second  range  corre- 
spond to  Oj  of  the  first  range. 

Then,  J^  and  K^  being  the  point  at  infinity  and  A^,  B^ 
being  any  points  of  the  first  range,  we  have 

(A  B  J K^  =  ^^'^^  •  ^^^1  =  1  •  ^i^i  =  A-^i, 
^    ^    ^  ^    ^^     B^J^'  B^K^        '  B^K^      A^K^ 

and  (^2^2'^2^2)  =  ^;  §22 

-t>2'^2 

and  since  {A^B^J^K^^  =  (A^B^J^K^),  we  have 

AiK^  .  A^J^  =  B^K^  .  ^2^2  =  .  .  .  =  OyK^  •  O^J^. 
Hence 

(Oi^i  -  Oi^i)  (Oi^2  -  ^1^2)  -  ^1^1  •  (C>i^2  -  0^0^  =  0, 
and  Oi^i  .  OjJg  -  O^A^  •  O^J^  -  O^K^  .  O^A^ 

-f  Oj^i  .  Oi^2  -  ^\^\  '  O1J2  +  O^K^  '0^0^  =  0. 
If  now  A^  and  A^  should  coincide  at  A,   a  self-corre- 
sponding point,  then,  since  O^J^  =  —  O^K^,   0^  being  the 
midpoint  of  J^Ki,  it  would  follow  that 

a;A^  =  -o,K, .  O1O2  =  0,J^  .  0,0^, 

Then  OiJ^  and  0^0^  would  agree  in  sign;  that  is,  Oj 
would  not  lie  between  J^  and  0^. 


CONSTANT  ANGLE  71 

On  the  other  hand,  if  0^  does  not  lie  between  0^  and 
Jg,  a  point  A,  related  to  0^,  0^,  J^  as  above,  may  be  found 
and  will  be  self-corresponding.  Hence,  when  there  is  no  self- 
corresponding  point,  Oj  must  be  within  the  segment  O^J^. 

Let  -STj,  Jj'  ^v  ^2  ^  indicated  on  the  base.  Erect  at 
Oj  the  perpendicular  O^F  meeting  at  P  the  semicircle 
whose  diameter  is  O^J^. 

Through  the  point  F  draw  a  line  parallel  to  -p. 


O2    O, 


Then  angle  O^FJ^  =  90°,  and  angles  K<^PK^  (acute), 
J^FJ^,  and  O^FO^  are  equal.  But  the  triads  K^J^^O^  and 
K^J^O^  determine  the  projectivity  of  the  ranges. 

Let  -4j,  A^  be  any  pair  of  corresponding  points. 

The  projectors  from  F  of  the  "ranges  K^O^J^A^  and 
^jOjJj^i  are  pencils  which  by  the  rotation  of  the  second 
through  the  common  value  6  of  the  three  angles  K^FK^, 
J^FJy,  O^FO^  could  be  made  to  have  three  common  lines 
while  the  projectivity  would  not  be  destroyed. 

Then  all  the  corresponding  lines  would  be  made  to 
coincide ;  and  by  the  rotation  of  FA^  through  the  angle  d 
it  would  be  brought  into  the  position  FA^ 

Hence  angle  A^FA^  =  0,  and  the  theorem  follows. 

In  Case  3  of  §  58  the  existence  of  superposed  projective  ranges 
having  no  self-corresponding  point  was  established  by  means  of  an 
example.  In  this  example  the  two  ranges  could  have  been  generated 
simultaneously  by  the  intersection  of  their  base  with  the  arms  of  an 
angle  of  constant  size  rotating  about  its  vertex.  It  has  just  been 
established  that  every  pair  of  such  superposed  projective  ranges  can 
be  generated  in  this  way. 


72  SUPERPOSED  PROJECTIVE  FORMS 

Theorem.  Involution  of  elements 

64.  If  the  projectivity  between  two  superposed  projective 
one-dimensional  forms  is  such  that  when  any  one  element  is 
taken  as  belonging  to  the  first  form  that  element  has  the  same 
corresponding  element  as  it  has  when  it  is  taken  as  belonging 
to  the  second  form^  then  every  element  has  this  property. 

Proof.  We  shall  consider  the  theorem  for  ranges  only, 
the  proof  being  similar  for  other  forms. 

Between  two  triads  A^B^C^  and  A^B^C^  that  lie  on  a 
base  p  there  exists  a  projectivity  which  determines  two 
superposed  projective  ranges  on  this  bsise.  In  connec- 
tion with  this  projectivity  every  point  of  the  line  p  may 
be  given  two  names.  Thus,  a  point  might  be  L^  and  R^. 
Moreover,  the  original  triads  A^B^C^  and  A^B^C^  may  have 
some  points  in  common. 

Suppose  now  that  B^  is  taken  to  be  the  same  as  A^^  and 
B^  the  same  as  A^.  Let  a  point  Z>j  be  taken  to  be  the 
same  as  C^.  The  theorem  is  then  proved  if  we  can  show 
that  i>2  coincides  with  Cj. 

From  the  conditions  of  the  case 

iA,A^C,C^^  =  (A,B,C,D,)  =  (A^B^C^D^)  =  (A^A.C^D^) ; 
whence  A^C^ :  A^C^  =  A^D^  :  A^D^, 

and  D^  coincides  with  Cj. 

65.  Involution.  When  two  superposed  prime  forms  are 
connected  by  a  projectivity  such  as  that  described  in  the 
above  theorem,  they  are  said  to  form  an  involution. 

Elements  of  an  involution  which  correspond  to  each 
other  are  said  to  be  conjugate. 

The  projectivity  is  also  called  an  involution. 

66.  Corollary.  Every  projection  and  every  section  of 
an  involution  of  elements  yields  an  involution. 


INVOLUTION  OF  ELEMENTS  73 

Theorem.  Anharmonic  Ratios  in  Hyperbolic  Involution 

67.  In  a  hyperbolic  involution  the  anharmonic  ratio  of  the 
two  self -corresponding  elements  and  any  pair  of  correspond- 
ing elements  is  —  1. 

Proof.  If  X,  Y  are  the  self-corresponding  points  and 
^1,  A^  are  corresponding  points  of  a  point  involution,  then 

Hence       (^XYA^A^y^l,   and   (^XYA^A^^  =  -1. 

(X  YA^A^)  cannot  be  equal  to  1,  for  X,Y,A.^,A^  are  distinct  points. 

The  classification  of  projectivities  into  hyperbolic,  parabolic,  and 
elliptic  applies  to  the  involutions,  and  it  is  easy  to  prove  that  invo- 
lutions of  all  these  classes  exist.  Hence  §§  58-61  apply  in  the  case 
of  involutions.  In  particular,  the  value  of  the  anharmonic  ratio 
mentioned  in  §  60  has  been  determined  in  the  above  theorem. 

The  converse  of  this  theorem  is  easily  proved;  and  hence, 
when  any  two  elements  of  a  range  or  pencil  are  given,  it  becomes 
easy  to  determine  any  desired  number  of  pairs  of  corresponding 
elements  of  an  involution  of  which  the  two  given  elements  shall 
be  self-corresponding.  A  similar  remark  may  be  made  regarding 
the  more  general  case  of  §  60. 

68.  Corollary  1.  In  a  hyperbolic  point  involution  the 
point  at  infinity,  if  not  self-corresponding,  corresponds  to  a 
point  midway  between  the  self -corresponding  points. 

69.  Corollary  2.  In  a  hyperbolic  point  involution,  if 
the  point  at  infinity  is  a  self-corresponding  point,  the  other 
self -corresponding  point  bisects  the  line  joining  every  pair  of 
corresponding  points. 

70.  Corollary  3.  iw  a  hyperbolic  involution  of  lines 
(or  planes^,  the  line  (or  plane^  which  bisects  the  angle  between 
the  self-corresponding  lines  (or  planes^  has  for  its  correspond- 
ing line  (or  plane^  the  one  which  is  perpendicular  to  it. 


74  SUPERPOSED  PROJECTIVE  FORMS 

71.  Involution  Determined.  An  involution  is  determined 
whenever  enough  is  known  to  establish  two  determining 
triads  of  the  projectivity.  Consequently  the  following 
data  are  sufficient  for  this  purpose: 

1.  Two  pairs  of  corresponding  elements. 

2.  One  self-corresponding  element  and  a  pair  of  corre- 
sponding elements. 

3.  Two  self-corresponding  elements. 

Considering  the  argument  for  ranges  only,  the  other 
cases  admitting  of  similar  treatment,  we  see  that  the  fol- 
lowing triads  determine  projectivities : 

In  Case  1  the  triads  A^A^B^,  A^A^B^,  where  A^,  A^ ;  B^,  B^ 
are  corresponding  pairs  of  points. 

In  Case  2  the  triads  XA^A^,  XA^A^,  where  X  is  a  self- 
corresponding  point  and  -4j,  A^  are  a  pair  of  correspond- 
ing points. 

In  Case  3  the  triads  XYA^,  XYA^,  where  X,  Y  are  self- 
corresponding  points  and  A^,  A^  are  a  pair  of  harmonic 
conjugates  with  respect  to  them. 

72.  Center  of  Involution.  Let  the  pairs  of  points  J,  0 ; 
-(4 J,  A^ ;  -Bj,  B^  (J  being  the  point  at  infinity)  be  corre- 
sponding. 

Then  (^^  JS^  0J)  =  {A^B^J  0) ; 

whence  —^  =  — ^ , 

B^O     A^O 

and  OAi  •  OA^  =  OB^  .  OB^  =  .... 

Then  in  a  point  involution  the  point  corresponding  to 
the  point  at  infinity  is  such  that  the  product  of  its  dis- 
tances from  any  pair  of  corresponding  points  is  independent 
of  the  choice  of  the  pair.  This  point  is  called  the  center 
of  the  im^olution. 


POINT  INVOLUTION  75 

73.  Two  Cases  of  Point  Involution.  A  further  examina- 
tion of  the  relations  just  found  suggests  two  possible  cases : 

1.  The  product  OA^  •  OA^  may  be  negative. 

2.  The  product  OA^  •  OA^  may  be  positive. 

In  Case  1  no  self-corresponding  point  X  can  exist ;  for 
if  it  did  we  should  have  OX^  negative,  which  is  impossible 
for  real  values  of  OX.    The  involution  is  therefore  elliptic. 

Also,  from  the  relation  OA^^  •  OA^  =  OB^  •  OB^,  if  each 
product  is  negative,  it  follows  that  the  point  0  separates 
every  pair  of  corresponding  points.  Moreover,  if  OA^  is 
longer  than  OB^,  it  is  evident  that  OA^  is  shorter  than  OB^, 
and  so  any  two  pairs  of  corresponding  points,  as  A^,  A^; 
i?j,  i?2'  mutually  separate  each  other. 

In  Case  2  by  similar  reasoning  we  establish  that  the 
involution  is  hyperbolic  with  the  self-corresponding  points 
equidistant  from  0,  that  any  two  corresponding  points  lie 
in  the  same  direction  from  O,  and  that  no  two  pairs  of 
corresponding  points  mutually  separate  each  other. 

These  metric  properties  are  sometimes  used  to  define  elliptic  and 
hyperbolic  point  involutions. 

Exercise  19.    Point  Involutions 

1.  Choosing  two  pairs  of  corresponding  points  that  will 
determine  an  elliptic  involution,  find  by  construction  a  third 
pair  of  corresponding  points  and  also  find  the  center. 

2.  Solve  Ex.  1  for  a  hyperbolic  involution. 

3.  Find  a  pair  of  corresponding  points  of  an  involution  of 
which  A'  (a  given  point)  and  /  (the  point  at  infinity)  are  the 
self-corresponding  points. 

4.  Given  one  self-corresponding  point  of  an  involution  and 
a  pair  of  corresponding  points,  find  by  construction  the  other 
self-corresponding  point. 


76  SUPERPOSED  PROJECTIVE  FORMS 

Theorem.  Line  Involution 

74.  Every  involution  of  lines  has  one  pair  of  correspond- 
ing lines  at  right  angles ;  if  it  has  two  pairs  at  right  angles^ 
all  its  pairs  are  at  right  angles. 


Proof.  Let  aj,  a^ ;  ftj,  hc^  be  pairs  of  corresponding  lines 
of  an  elliptic  involution  on  a  base  P.  If  a^,  a^  or  5j,  h^ 
are  at  right  angles,  the  first  part  needs  no  proof. 

If  neither  of  these  pairs  of  lines  is  at  right  angles,  cut 
the  four  lines  by  any  line  p  in  the  points  ^j,  A^^  B^,  B^. 

Describe  the  circles  PA^A^  and  PB^B^  meeting  again 
in  Q.  heiPQcnt  p  in  O,  and  let  the  perpendicular  bisector 
of  PQ  cut  p  in  V. 

Describe  the  circle  whose  center  is  V  and  whose  radius 
is  VPt  or  VQ,  and  let  it  cut  p  in  Cj  and  C^, 

Then        OC^  .  OC^  =  OP'OQ 

=  OA^ .  0^2  =  ^^1  •  OB^. 

Hence  the  lines  c^,  or  PC^,  and  c^,  or  PC^,  belong  to  the 
original  involution.  Furthermore,  they  are  at  right  angles, 
because  C^PC^  is  a  semicircle. 

For  the  case  of  a  hyperbolic  involution  see  §  70. 


LINE  INVOLUTION  77 

Again,  if  two  pairs  of  corresponding  lines  are  at  right 
angles,  these  pairs  separate  each  other,  and  the  involution 
is  elliptic. 

Suppose  now  that  the  lines  a^,  a^  and  also  the  lines  6j,  h^ 
are  at  right  angles.  Then  the  segments  PA^A^  and  PB^B^ 
are  semicircles,  and  their  common  chord  PQ  is  at  right 
angles  to  p  at  0.  Now  if  c^,  c^-,  any  other  pair  of  corre- 
sponding lines  of  the  involution,  cut  p  at  Cj,  Cg,  then 

OCi  .  OC2  =  OP  .  OQ, 

and  the  segment  C^PC^  is  a  semicircle. 

Accordingly,  <?j,  c^,  any  two  corresponding  lines  of  the 
involution,  are  at  right  angles. 

Exercise  20.   Review 

1.  On  a  given  base  p,  if  each  of  two  ranges  is  projective 
with  a  third  range  in  such  a  way  that  X  and  Y  are  self- 
corresponding  points  for  both  projectivities,  then  these  two 
ranges  are  projective  with  each  other,  and  X  and  Y  are  self- 
corresponding  points  in  the  projectivity  connecting  them. 

2.  In  Ex.  1,  if  the  anharmonic  ratios  (§  60)  associated  with 
the  two  projectivities  are  r^  and  r^,  find  the  anharmonic  ratio 
associated  with  the  third  projectivity. 

3.  Interpret  the  results  of  Exs.  1  and  2  when  the  first  two 
projectivities  are  involutions. 

4.  A  projectivity  between  superposed  forms  is  determined 
if  two  self-corresponding  elements  and  the  anharmonic  ratio 
of  these  elements  and  two  corresponding  elements  are  given. 

5.  Given  two  self-corresponding  points  X  and  Y  of  two 
superposed  projective  i-anges,  find  a  third  range  on  the  same 
base  between  which  and  each  of  the  other  two  ranges  there 
exists  a  projectivity  for  which  X  and  Y  are  self-corresponding 
points.    Consider  the  number  of  solutions. 


78  SUPERPOSED  PROJECTIVE  FORMS 

6.  A  point  involution  is  completely  determined  by  its 
center  and  one  self-corresponding  point. 

7.  A  point  involution  is  completely  determined  by  its  center 
and  a  pair  of  corresponding  points. 

8.  The  circles  which  pass  through  two  given  fixed  points 
determine,  on  any  line  which  cuts  them,  corresponding  points 
of  an  involution. 

9.  If  .1,  B,  C,  D  are  fixed  coplanar  but  noncollinear  points 
and  a  point  0  moves  in  their  plane  in  such  a  way  that  the 
anharmonic  ratio  of  the  i^encil  0{ABCD)  is  a  given  constant  k, 
the  lines  OC  and  OD  meet  the  line  AB  in  corresponding  points 
of  two  superposed  projective  ranges. 

10.  Consider  Ex.  9  for  the  case  in  which  ^  =  —  1. 

11.  In  Ex.  9  the  path  of  O  intersects  the  line  AB  in  A  and  B 
and  in  no  other  points,  and  hence  it  passes  through  all  four 
of  the  points  A,  B,  C,  D. 

12.  Every  line  through  one  of  the  points  A,  B,  C,  Z)  in  Ex.  9 
cuts  the  path  of  0  in  one  and  only  one  additional  point. 

13.  In  Ex.  12  find  the  other  point  in  which  the  path  of  O  is 
cut  by  any  given  line  through  A. 

14.  In  Ex.  9,  assuming  that  as  a  point  moving  along  a  curve 
approaches  a  fixed  point  the  secant  through  the  two  points  ap- 
I)roaches  the  tangent  to  the  curve  at  the  fixed  point,  construct 
the  tangent  to  the  path  of  O  at  the  point  A. 

15.  If  ylj,  A^,  A^  are  noncollinear  points,  a^,  a^,  a^  noncon- 
current  lines,  P^  a  point  on  a^,  P^A^  and  a^  intersect  in  P^, 
P^A^  and  Og  in  Pg,  and  P^A^  and  a^  in  P[,  then,  as  P^  and  P[ 
move  along  a^,  they  trace  two  superposed  projective  ranges. 

16.  In  Ex.  15  under  what  circumstances  (if  any)  do  the 
lines  Pj^i,  P^A^,  and  P^A^  form  a  triangle  with  its  vertices 
on  the  three  given  lines  ? 

17.  What  modifications  of  the  data  in  Ex.  15  render  it  cer- 
tain that  in  all  positions  the  lines  P^A^,  P^^A,^,  and  P^A^  form  a 
triangle  with  its  vertices  on  the  same  three  given  lines  ? 


PAKT  11.   APPLICATIONS 
CHAPTER  VIII 

PKOJECTIVELY  GENERATED  FIGURES 

75.  Statement  of  the  General  Problem.  Application  of 
the  properties  of  prime  forms  will  now  be  made  to  the 
study  of  a  problem  which  is  connected  with  a  somewhat 
wide  range  of  topics  in  geometry.  This  problem,  to  the 
discussion  of  which  the  remainder  of  the  book  is  devoted, 
may  be  stated  in  these  words: 

To  determine  the  character  of  all  geometric  configurations 
whose  generating  elements  are  determined  hy  corresponding 
elements  of  two  projective  one-dimensional  prime  forms. 

The  problem  divides  naturally  into  a  number  of  cases 
according  as  the  two  projective  one-dimensional  prime 
forms  are  any  one  of  the  following  pairs: 

1.  Two  ranges,  considered  in  §§  84  and  85. 

2.  Two  flat  pencils,  considered  in  §§  86-89. 

3.  Two  axial  pencils,  considered  in  §§  90  and  95. 

4.  A  range  and  a  flat  pencil,  considered  in  §  96. 

5.  A  range  and  an  axial  pencil,  considered  in  §  97. 

6.  A  flat  pencil  and  an  axial  pencil,  considered  in  §  98. 

76.  Locus  of  a  Point.  If  a  point  moves  in  space  subject 
to  a  given  law,  the  figure  consisting  of  all  the  points  with 
which  the  moving  point  may  coincide,  and  of  no  others, 
is  called  the  locus  of  the  point. 

79 


80         PROJECTIVELY  GENERATED  FIGURES 

77.  Envelope  of  a  Plane.  If  a  plane  moves  in  space 
subject  to  a  given  law,  the  figure  which  is  tangent  to  all 
the  planes  with  which  the  moving  plane  may  coincide,  and 
to  no  others,  is  called  the  envelope  of  the  plane. 

78.  Envelope  of  a  Line.  If  a  line  moves  in  a  plane  sub- 
ject to  a  given  law,  the  figure  which  is  tangent  to  all  the 
lines  with  which  the  moving  line  may  coincide,  and  to  no 
others,  is  called  the  (^plane}  envelope  of  the  line. 

79.  Generation  of  a  Figure  by  a  Line.  If  a  line  moves 
in  space  subject  to  a  given  law,  the  figure  consisting  of  all 
the  lines  with  which  the  moving  line  may  coincide,  and  of 
no  others,  is  said  to  be  generated  by  the  moving  line. 

80.  Order  of  a  Figure.  The  greatest  number  of  points 
of  a  figure  that  lie  on  a  line  which  is  not  entirely  in  the 
figure  is  called  the  order  of  the  figure. 

Thus  a  circle  may  be  met  by  a  line  in  two,  one,  or  no  points. 
Consequently  the  order  of  the  circle  is  two.  Similarly,  the  order  of  a 
straight  line  is  one.  The  order  of  a  plane  is  also  one,  while  that  of 
a  sphere  is  two. 

81.  Class  of  a  Figure  in  Space.  The  greatest  number  of 
tangent  planes  of  a  figure  in  space  which  pass  tlirough  a 
line  that  does  not  have  all  the  planes  through  it  tangent 
to  the  figure  is  called  the  class  of  the  figure. 

Thus  a  sphere  may  have  tangent  to  it  two,  one,  or  no  planes  which 
pass  through  a  straight  line,  and  hence  the  sphere  is  of  class  two. 

82.  Class  of  a  Figure  in  a  Plane.  The  greatest  number 
of  tangent  lines  which  can  be  drawn  to  a  plane  figure  from 
any  point  in  its  plane  is  called  the  class  of  the  figure. 

Thus,  of  the  lines  in  a  plane  which  pass  through  a  given  point 
two,  one,  or  none  may  be  tangent  to  a  g^ven  circle,  and  hence  the 
circle  is  of  class  two. 


DEFINITIONS  81 

83.  Dual  Elements.  From  the  point  of  view  of  the  Prin- 
ciple of  Duality  the  following  are  corresponding  elements : 

1.  In  geometry  of  three  dimensions,  a  point  on  a  figure 
and  a  plane  tangent  to  a  figure. 

It  can  also  be  shown  that  a  line  on  a  figure  is  self-dual. 

2.  In  geometry  of  the  plane,  a  point  on  a  figure  and  a  line 
tangent  to  a  figure. 

3.  In  geometry  of  the  bundle,  a  line  on  a  figure  and  biplane 
tangent  to  a  figure. 

Exercise  21.    Preliminary  Definitions 

1.  What  is  the  envelope  of  a  system  of  tangents  to  a  given 
circle  ?    What  is  the  dual  of  a  circumscribed  polygon  ? 

2.  What  is  the  order  and  what  is  the  class  of  the  projector 
of  a  circle  from  a  point  not  in  its  plane  ? 

The  student  may  consult  the  chapters  on  higher  plane  curves  in  texts 
on  elementary  analytic  geometry,  such  as  the  one  in  this  series,  and  deter- 
mine the  orders  and  the  classes  of  the  curves  considered  there. 

3.  In  Ex.  2  what  is  the  order  and  what  is  the  class  of  any 
plane  section  of  the  figure  ? 

4.  Find  the  surface  generated  by  a  line  so  moving  as  to  be 
constantly  parallel  to  and  at  a  given  distance  from  a  given  line, 
and  state  the  order  and  the  class  of  this  surface. 

5.  In  Ex.  4  consider  the  various  plane  sections  of  the  sur- 
face, stating  the  order  and  the  class  of  each. 

6.  Find  the  locus  in  space  of  a  point  which  so  moves  as  to 
be  constantly  at  a  given  distance  from  the  nearest  point  of  a 
given  line  segment,  stating  the  order  and  the  class  of  the  locus. 

7.  Find  the  order  and  the  class  of  the  plane  sections  of  the 
figure  obtained  in  Ex.  6. 

8.  Find  the  envelope  of  a  plane  which  so  moves  as  to  be 
constantly  at  a  given  distance  from  the  nearest  point  of  a 
given  line  segment,  stating  the  order  and  the  class  of  the  locus. 


82        PROJECTIVELY  GENERATED  FIGURES 

Theorem.  Ranges  with  a  common  Element 

84.  The  envelope  of  the  line  which  so  moves  as  always  to 
contain  corresponding  points  of  tivo  coplanar  projective  ranges 
is  of  the  second  class,  unless  the  ranges  are  superposed  and 
without  a  self-corresponding  point,  in  which  case  the  envelope 
is  the  common  base.  If  the  ranges  are  perspective,  the  enve- 
lope consists  of  tivo  points,  one  of  tvhich  is  common  to  the 
ranges.  If  the  ranges  are  not  perspective  and  not  superposed, 
the  envelope  is  tangent  to  the  base  of  each  range  at  that  point 
of  it  which  corresponds  to  the  point  common  to  the  ranges 
when  this  common  point  is  regarded  as  belonging  to  the  other 
range. 

Proof.  The  two  projective  ranges  referred  to  in  §  75 
may  or  may  not  have  one  common  element,  and  in  this 
theorem  we  consider  two  ranges  having  such  an  element. 
In  this  case  the  ranges  must  evidently  be  coplanar. 

The  element  determmed  by  two  corresponding  points  of 
the  ranges  is  a  straight  line,  and  hence  in  this  case  the 
problem  is  that  of  determining  the  envelope  of  a  line 
which  so  moves  as  always  to  contain  two  corresponding 
points  of  two  coplanar  projective  ranges. 

These  ranges  may  be  (1)  superposed ;  (2)  not  superposed 
but  perspective ;  (3)  neither  superposed  nor  perspective. 

1.  Let  the  ranges  be  superposed. 

Then  all  pairs  of  corresponding  but  not  self-corresponding 
points  determine  the  common  base  of  the  ranges ;  and  the 
lines  through  any  self-corresponding  point  are  infinitely 
many.  The  only  figure  that  has  all  these  lines  and  no  others 
as  tangents  consists  of  the  one  or  two  self -corresponding 
points  or,  if  there  is  no  self-corresponding  point,  consists 
of  the  common  base  of  the  ranges. 


RANGES  WITH  A  COMMON  ELEMENT        83 


2.  Let  the  ranges  he  not  superposed  hut  perspective. 

Then  all  lines  determined  by  distinct  corresponding 
points  pass  through  the  center  of  perspective,  and  every 
line  through  the  point  common  to  the  ranges  joins  that 
point  to  its  corresponding  point,  that  is,  to  itself.  Hence 
the  envelope  consists  of  two  points ;  namely,  the  center  of 
perspective  and  the  point  common  to  the  ranges. 

3.  Let  the  ranges  he  neither  superposed  nor  perspective. 
To  the  common  point  Xj,  or  Y^,  there  correspond  in  the 

ranges  two  points  X^  and  Fj.    Since  the  base  p^  of  the  first 
range  joins  Y^  to  F^,  the  enve-  p 

lope  is  tangent  to  p^ ;  similarly, 
it  is  tangent  to  p^. 

Consider  now  Fj  and  Fj,  two 
corresponding  points  nearly  co- 
incident with  Xj  and  Xj  respec- 
tively. The  line  v  joining  them 
is  nearly  coincident  with  p^.  If, 
now,  Fj  approaches  coincidence 
with  Xj,  V  and  Fj  approach  p^ 
and  Xg.  But  if  a  moving  tan- 
gent to  a  curve  approaches  a 
fixed  tangent  as  a  limiting  line,  the  intersection  of  these 
two  approaches  the  point  of  contact  of  the  fixed  tangent 
as  a  limiting  pomt.  Hence  in  this  case  Xg  is  the  point  of 
contact  of  p^  with  the  envelope.  Similarly,  Fj  is  the  point 
of  contact  of  py 

Finally,  the  class  of  the  envelope  is  two ;  for  two  tan- 
gents to  the  envelope,  namely,  p^  and  A^A^,  pass  through 
a  point  Ay  If  through  any  point  0  there  should  pass  three 
tangents  to  the  envelope,  0  would  be  a  center  of  perspec- 
tive for  the  ranges ;  but  the  ranges,  are  not  perspective. 


84         PROJECTIVELY  GENERATED  FIGURES 

Theorem,  ranges  with  no  common  element 

85.  The  lines  which  join  corresponding  points  of  two  pro- 
jective ranges  that  have  no  common  point  are  the  intersections 
of  corresponding  planes  of  two  projective  axial  pencils  which 
have  no  common  plane. 


Proof.  If  two  projective  ranges  have  no  common  ele- 
ment, their  bases  cannot  meet,  and  therefore  the  ranges 
cannot  be  coplanar. 

Let  A^B^C^  •  •  •  and  A^B^C^  •  •  •  be  projective  ranges  on 
the  bases  pi  and  p^  which  are  not  coplanar. 

Let  a^,  y8j,  7^,  •  •  •  be  the  planes  determined  by  the  line  pi 
and  the  points  A^,  B^,  Cg,  •  •  •  respectively,  and  let  a^,  jS^,  72?  •  •  • 
be  the  planes  determined  by  the  line  p^  and  the  points 
^1,  i?j,  Cj,  •  •  •  respectively.    Then  we  have 

axial  pencil  cc^^ff^ '  *  '  a  ^^"S^  ^2 ^2 ^2  *  *  * 
-range  ^1^1  Cj... 
-  axial  pencil  ^2^272  *  *  - 

Also  the  line  A^A^  is  the  intersection  of  the  planes  a^ 
and  a^.  Moreover,  if  the  axial  pencils  had  a  common 
plane,  the  ranges  along  their  axes  would  both  be  in  this 
plane.    But  this  is  contrary  to  hypothesis. 

Hence  the  proof  is  complete. 

As  a  consequence  of  this  theorem  the  discussion  of  the  figure 
which  is  generated  by  these  two  projective  ranges  may  be  deferred 
until  we  consider  the  figure  which  is  generated  by  projective  axial 
pencils  that  have  no  common  plane  (§  95). 


KANGES  AND  PENCILS  85 

Theorem,  coplanar  Flat  Pencils 

86.  The  locus  of  the  point  which  so  moves  as  always  to  be 
common  to  tivo  corresponding  lines  of  two  coplanar  projective 
flat  pencils  is  of  the  second  order,  unless  the  pencils  are  super- 
posed and  without  a  self-corresponding  line,  in  which  case  the 
locus  is  the  common  vertex.  If  the  pencils  are  perspective,  the 
locus  consists  of  two  straight  lines,  one  of  which  is  common  to 
the  pencils.  If  the  pencils  are  not  superposed,  the  locus  con- 
tains the  base  of  each  pencil  and  at  each  of  these  points  is 
tangent  to  the  line  of  the  corresponding  pencil  which  corre- 
sponds to  the  common  line  of  the  pencils  when  this  common 
line  is  regarded  as  belonging  to  the  other  pencil. 

Proof.  Two  flat  pencils  may  or  may  not  have  a  common 
base.  In  the  former  case  the  common  base  may  be  a  plane 
contaming  both  pencils  or  a  point  which  is  the  vertex  of 
both  pencils.  We  shall  now  deal  with  the  first  of  these 
subcases,  the  second  being  discussed  in  §  87. 

Essentially,  then,  the  theorem  involves  the  problem  of 
finding  the  locus  of  the  intersection  of  two  coplanar  pro- 
jective flat  pencils.  It  is  evident  that  these  pencils  may 
be  (1)  superposed ;  (2)  not  superposed  but  perspective ; 
(3)  neither  superposed  nor  perspective. 

1.  Let  the  pencils  be  superposed. 

In  this  case  the  flat  pencils  have  in  common  not  only 
their  planes  but  also  their  vertices.  The  points  common  to 
the  corresponding  lines  include  in  any  case  the  common 
vertex  and  also  include  all  points  of  any  self -corresponding 
lines  of  the  pencil.  Hence  in  this  case  the  locus  is  the  one 
or  two  self-corresponding  lines  of  the  pencils  or,  in  case 
there  is  no  self-corresponding  line,  the  common  point  of  all 
the  lines  of  the  pencil. 


86         PROJECTIVELY  GENERATED  FIGURES 

2.  Let  the  pencils  he  not  superposed  but  perspective. 

In  this  case  the  pencils  have  a  self-corresponding  line, 
all  the  points  of  which  are  in  the  locus.  In  addition  the 
intersections  of  pairs  of  correspondmg  lines  are  in  the 
locus.  But  these  intersections  are  on  a  straight  line,  and 
accordingly  the  locus  is  a  pair  of  straight  lines. 

3.  Let  the  pencils  he  neither  superposed  nor  perspective. 


Since  the  common  line  is  not  self-corresponding,  let  it 
be  called  x^  and  ^j*  ^^  the  two  pencils  it  has  corresponding 
to  it  the  lines  ^j  and  x^. 

The  base  J^  of  the  first  pencil,  being  the  intersection  of 
the  lines  t/^  and  i/^,  is  on  the  locus.  Similarly,  the  base  F^ 
of  the  second  pencil  is  on  the  locus. 

Consider  now  a  line  v^  that  is  nearly  coincident  with  a;j, 
and  the  corresponding  line  v^  that  is  nearly  coincident  with 
Zj.  The  point  Fof  the  locus  determined  by  Vj  and  v^  is  nearly 
coincident  with  i^.  If,  now,  the  point  V  approaches  F^  along 
the  locus,  the  line  Vg,  or  VJ^,  approaches  x^.  But  the  limit- 
ing position  of  a  secant  Vl^  as  V  approaches  ^  is  the  tan- 
gent to  the  locus  at  i^.  Hence  the  locus  is  tangent  to  the 
line  x^  at  I^.  Similarly,  it  follows  that  the  line  y^  is  tangent 
to  the  locus  at  J^. 


RANGES  AND  PENCILS  87 

Finally,  the  order  of  the  locus  is  two;  for  any  line  of 
the  first  pencil,  as  a^,  meets  the  locus  at  J^  and  at  A,  the 
intersection  of  a^  and  its  corresponding  line  a^.  Moreover, 
if  any  line  o  cuts  the  locus  in  three  points,  the  triads  of 
lines  of  the  pencils  which  would  intersect  in  these  points 
would  be  perspective,  and  so  would  the  complete  pencils. 
But  the  pencils  are  not  perspective.  The  theorem  is, 
therefore,  completely  proved. 

Exercise  22.    Ranges  and  Pencils 

1.  Two  fixed  lines  AO^B  and  CO^D  are  each  perpendicular 
to  0^0^,  and  a  moving  line  cuts  them  in  P^  and  P^  so  that  the 
ratio  O^P^ :  O^P^  is  a  constant.  Find  the  envelope  of  the  moving 
line.    Consider  the  case  in  which  O^P^  and  O^P^  are  equal. 

Compare  the  ranges  traced  by  P^  and  Pg. 

2.  Two  fixed  lines  intersect  at  right  angles  at  O,  and  a 
moving  line  cuts  them  at  equal  distances  from  O.  Find  the 
envelope  of  the  moving  line. 

3.  Examine  Ex.  1,  substituting  the  condition  that  O^P^  and 
O^P^  maintain  a  constant  difference. 

4.  Examine  Ex.  2,  substituting  the  condition  that  the  dis- 
tances from  0  maintain  a  constant  difference. 

5.  Examine  Ex.  2,  substituting  the  condition  that  one  dis- 
tance exceeds  a  given  multiple  of  the  other  by  a  fixed  amount. 

6.  Two  lines  revolve  at  the  same  angular  velocity  in  opposite 
senses  about  the  fixed  points  0^  and  0^  respectively.  Initially 
they  make  angles  of  90°  and  45°  respectively  with  the  line  0^0.^. 
Find  the  locus  of  their  intersection. 

7.  Consider  Ex.  6,  substituting  the  condition  that  initially 
the  lines  coincide. 

8.  Examine  Exs.  6  and  7  on  the  assumption  that  the  lines 
revolve  in  the  same  sense. 


88        PROJECTIVELY  GENERATED  FIGURES 

THEOREM.  Flat  Pencils  with  a  common  Vertex 
87.  TJie  envelope  of  the  plane,  which  so  moves  as  always  to 
contain  corresponding  lines  of  two  projective  fiat  pencils  that 
are  not  coplanar  hut  have  a  common  vertex,  is  of  the  second 
class.  If  the  flat  pencils  are  perspective,  the  envelope  consists 
of  ttco  straight  lines,  one  of  tvhich  is  common  to  the  pencils. 
If  the  pencils  are  not  perspective,  the  envelope  is  a  surface 
tangent  to  the  plane  of  each  fiat  pencil  along  the  line  which 
corresponds  to  the  common  line  of  the  pencil  when  this  cotnmon 
line  is  regarded  as  belonging  to  the  other  flat  pencil.  All  the 
planes  and  the  mrface  generated  pass,  through  the  common 
vertex  of  the  pencil. 


Proof.  Since  each  pair  of  corresponding  lines  of  the 
given  flat  pencils  determines  both  a  point  and  a  plane, 
we  are  concerned  with  the  problem  of  finding  the  aggre- 
gate of  elements,  either  points  or  planes,  determined  by 
corresponding  lines  of  two  projective  flat  pencils  which 
have  a  common  vertex.    The  first  of  these  cases  is  trivial. 

It  may  here  be  assumed  that  the  flat  pencils  are  not  coplanar,  as 
the  case  of  coplanar  pencils  has  just  been  discussed. 

From  the  point  of  view  of  loci  the  points  determined  by 
corresponding  lines  either  will  be  the  common  vertex  alone 
or,  if  the  common  line  happens  to  be  self-corresponding, 
will  be  this  common  line  itself. 


FLAT  PENCILS  89 

On  the  other  hand,  any  two  corresponding  lines  deter- 
mine a  plane  which  passes  through  the  common  vertex. 
If  we  cut  the  whole  configuration  by  a  plane  that  does  not 
pass  through  the  common  vertex,  we  obtain  two  pro- 
jective coplanar  ranges  and  the  lines  joining  corresponding 
points.  This  section  of  the  envelope  is  then  one  of  the 
figures  described  in  §  84,  Consequently  the  envelope 
sought  is  the  projector,  from  the  common  base  of  the  two 
flat  pencils,  of  one  of  these  figures.  If  the  flat  pencils  are 
perspective,  this  projector  consists  of  two  straight  lines 
through  the  common  vertex.  If  the  flat  pencils  are  not 
perspective,  the  projector  is  a  conic  surface  and  is  tangent 
to  the  plane  of  each  flat  pencil.  This  is  evident  from  the 
fact  that  the  plane  of  either  pencil  is  determined  by  the 
common  line  of  the  pencils  and  x^  that  one  of  its  own  lines 
which  corresponds  to  the  common  line.  The  plane  of  this 
pencil  has  the  line  x  in  common  with  the  envelope. 

Through  the  common  line  of  the  pencils  there  pass  the 
two  planes  of  the  pencils,  and  these  are  tangent  to  the 
envelope.  If  tlu-ough  any  line  there  should  pass  more  than 
two  tangent  planes,  the  given  pencils  would  be  perspective. 

88.  Flat  Pencils  having  no  Common  Base.  The  discus- 
sion of  the  case  of  projective  flat  pencils  that  have  a  com- 
mon line  but  are  not  coplanar  and  do  not  have  a  common 
vertex  can  be  given  quickly.  No  intersection  of  corre- 
sponding lines  can  occur  except  on  the  common  line.  The 
common  line  may  be  self-corresponding,  in  which  case  any 
point  on  it  is  common  to  corresponding  lines,  and  any 
plane  through  it  contains  corresponding  lines.  If  the 
common  line  is  not  self-corresponding,  there  is  in  each 
pencil  a  line  corresponding  to  it.  With  these  lines  it  deter- 
mines two  points,  the  vertices  of  the  pencils,  and  two  planes, 
the  planes  of  the  pencils. 


90         PROJECTIVELY  GENERATED  FIGURES 

89.  Projective  Flat  Pencils  having  no  Common  Element. 
Figures  generated  by  means  of  projective  flat  pencils  hav- 
ing no  common  line  are  so  simple  as  not  to  demand  special 
study.  They  will  be  noticed  in  passing,  but  no  formal 
theorem  regarding  them  need  be  stated. 

Since  pencils  which  lie  in  the  same  plane  or  in  differ- 
ent planes  that  intersect  in  a  line  of  the  pencils  have  a 
common  element,  it  follows  that  the  pencils  under  consider- 
ation lie  in  planes  whose  intersection  does  not  pass  through 
the  vertex  of  either  pencil.  Upon  this  line  the  pencils 
determine  ranges  which  may  be  either  identical  or  distinct. 

If  the  ranges  are  identical,  their  common  base  is  the 
locus  of  the  intersections  of  corresponding  lines  of  the  pen- 
cils. Likewise  each  pair  of  corresponding  lines  determines 
a  plane  through  the  line  joining  the  bases  of  the  flat  pencils ; 
and  the  envelope  of  these  planes  is  this  line.  Hence  the 
figure  determined  is  either  the  line  determined  by  the  ver- 
tices of  the  pencils  or  the  line  determined  by  the  planes 
of  the  pencils,  according  to  the  point  of  view. 

If  the  superposed  ranges  mentioned  above  are  not  iden- 
tical, the  only  pairs  of  corresponding  lines  of  the  flat  pen- 
cils which  determine  elements  are  those  which  meet  in  the 
self-corresponding  points  of  the  superposed  ranges.  These 
determine  two,  one,  or  no  points  on  the  line  common  to  the 
planes  of  the  pencils,  or  two,  one,  or  no  planes  through 
the  line  joining  the  vertices.  Hence,  from  the  point  of  view 
of  loci,  the  figure  generated  by  means  of  the  projective  flat 
pencils  consists  of  two,  one,  or  no  points ;  and  from  the 
point  of  view  of  envelopes,  the  figure  consists  of  two,  one, 
or  no  planes.  Though  neither  of  the  configurations  obtained 
has  any  special  interest  for  us,  it  is  clear  that  they  conform 
in  a  general  way  to  the  type  of  figures  which  we  obtain 
in  the  other  cases. 


FLAT  AND  AXIAL  PENCILS  91 

Theorem.  Axial  Pencils  with  a  common  plane 

90.  The  surface  generated  by  the  line  which  so  moves  as 
always  to  be  contained  in  corresponding  planes  of  two  axial 
pencils  that  have  a  common  plane  is  of  the  second  order 
unless  the  axial  pencils  are  superposed  and  are  without  self- 
corresponding  planes,  in  ivhich  case  the  surface  degenerates 
into  the  common  axis  of  the  pencils.  If  the  axial  pencils  are 
perspective,  the  surface  consists  of  two  planes,  one  of  which 
is  common  to  the  pencils.  If  the  pencils  are  neither  perspective 
nor  superposed,  the  surface  contains  the  axis  of  each  pencil 
and  along  each  of  these  axes  is  tangent  to  the  plane  which 
corresponds  to  the  common  plane  of  the  pencil  when  this  com- 
mon plane  is  regarded  as  belonging  to  the  other  pencil.  The 
generating  line  continually  passes  through  the  intersections  of 
the  axes  of  the  pencils. 

Proof.  Two  projective  axial  pencils  may  have  or  may 
not  have  a  common  element.  In  this  theorem  we  con- 
sider only  the  former  case.  Evidently  the  axes  of  the  two 
pencils  are  coplanar.  Then  the  pencils  may  be  (1)  super- 
posed ;  (2)  not  superposed  but  perspective ;  (3)  neither 
superposed  nor  perspective. 

1.  Let  the  axial  pencils  be  superposed. 

Then  all  pairs  of  corresponding  planes  intersect  in  the 
axis,  but  any  line  in  a  self-corresponding  plane  may  be 
regarded  as  common  to  two  coincident  corresponding 
planes.  The  surface  generated  by  the  lines  common  to 
corresponding  planes  consists,  therefore,  of  one  or  two 
self-corresponding  planes  if  there  are  such  planes  or,  if 
there  are  no  such  planes,  this  surface  degenerates  into 
a  line,  the  common  axis  of  the  pencil.  This  last  case  is 
of  minor  importance  only. 


92        PROJECTIVELY  GENERATED  FIGURES 

2.  Let  the  axial  pencils  he  not  superposed  hut  perspective. 
The  axes  of  the  pencils  intersect  in  a  point  0  which 

lies  in  every  plane  of  both  pencils.  Then  the  Ime  deter- 
mined by  any  pair  of  corresponding  planes  passes  through 
this  point.  Moreover,  if  a  plane  is  passed  so  as  not  to  con- 
tain this  point,  it  cuts  the  axial  pencils  in  coplanar  per- 
spective flat  pencils,  and  the  locus  of  the  intersections  of 
corresponding  lines  of  these  pencils  is  two  straight  lines, 
one  of  which  is  the  line  through  the  bases  of  the  flat  pencils 
(§  86).  Consequently  the  surface  generated  by  the  inter- 
sections of  corresponding  planes  of  the  axial  pencils  is  the 
projector  of  the  two  straight  lines  from  the  point  0 ;  that 
is,  the  surface  is  two  planes,  one  of  which  is  the  common 
plane  of  the  axial  pencil. 

3.  Let  the  axial  pencils  he  neither  superposed  nor  perspective. 

P2\,Pl 


As  in  the  previous  case,  the  axes  jOj  and  p^  intersect  in 
a  pomt  0  through  which  passes  every  line  determined  by 
corresponding  planes  of  the  axial  pencils.  A  plane  which 
does  not  pass  through  0  cuts  the  axial  pencils  in  coplanar 
projective  but  not  perspective  flat  pencils  whose  bases  are 
-Pj  and  Pj,  the  locus  of  whose  intersections  is  described  by 
§  86.  The  surface  generated  by  the  lines  common  to  cor- 
responding planes  of  the  axial  pencils  is  the  projector  from 
the  point  0  of  the  locus  just  mentioned. 


RULED  SURFACES 


93 


91.  Regulus.  Any  three  straight  lines,  no  two  of  which 
are  coplanar,  are  met  by  infinitely  many  straight  lines 
which,  taken  as  an  aggregate,  are  said  to  form  a  regulus. 

The  three  given  Imes  are  called  the  directrices  of  the 
regulus,  and  the  lines  which  meet  the  directrices  are  called 
the  generators  of  the  regulus. 


Kegulus 


QuADRic  Surface 


Skew  Quadric  Ruled  Surfaces 

92.  Quadric  Surface.  The  aggregate  of  the  points  of  the 
lines  of  a  regulus  constitute  a  surface  called  a  quadric 
surface,  and  the  generators  of  the  regulus  are  also  called 
the  generators  of  this  surface. 

There  are  quadric  surfaces  which  are  not  constituted  in  this  way. 

93.  Ruled  Surface.  A  surface  generated  by  the  move- 
ment of  a  straight  line  is  called  a  ruled  surface. 

94.  Skew  Ruled  Surface.  A  ruled  surface  in  which  no 
two  consecutive  generators  intersect  is  called  a  shew  ruled 
surface. 

For  a  full  discussion  of  skew  ruled  surfaces  see  §  187. 


94        PROJECTIVELY  GENERATED  FIGURES 

Theorem.  Axial  pencils  with  No  common  element 

95.  The  lines  of  intersection  of  two  projective  axial  pencils 
which  have  no  common  element  generate  a  skew  ruled  surface 
of  the  second  order  in  which  lie  the  bases  of  the  pencil.  Every 
section  of  this  surface  by  a  plane  through  a  generating  line 
is  a  pair  of  straight  lines.  All  other  plane  sections  are  curves 
of  the  second  order. 

Proof.  The  proof  may  be  divided  into  five  parts : 

1.  The  ruled  surface  is  skew. 

Each  generator  intersects  each  axis.  If  two  generators 
intersect,  they  determine  a  plane  which  contains  two  points 
and  therefore  all  points  of  each  axis.  This  plane  must 
then  be  a  common  element  of  the  pencils,  which  is  con- 
trary to  hypothesis.  Hence  no  two  generators  intersect, 
and  so  the  surface  is  skew. 

2.  The  bases  of  the  two  axial  pencils  lie  in  the  surface. 

Through  any  point  A  of  the  base  of  either  axial  pencil 
there  pass  all  the  planes  of  that  pencil  and  one  plane  a 
of  the  other.  Hence  A  is  on  the  line  determined  by  the 
plane  a  and  its  corresponding  plane.  Hence  the  surface 
contains  every  point  of  the  base  of  either  pencil. 

3.  Every  section  of  the  surface  by  a  plane  which  does  not 
pass  through  a  generating  lirie  is  a  curve  of  the  second  order. 

Any  plane  tt  which  does  not  contain  a  generating  line 
cuts  the  axial  pencils  in  two  projective  flat  pencils.  If 
these  flat  pencils  were  perspective,  their  common  line  would 
be  self-corresponding  and  the  plane  tt  would  cut  two  planes 
of  the  axial  pencils  in  their  common  line,  that  is,  in  a  gen- 
erating line;  and  this  is  contrary  to  hypothesis.  Then 
(§  86)  the  section  is  a  curve  of  the  second  order. 


AXIAL  PENCILS  96 

4.  Every  section  of  the  surface  hy  a  plane  through  a  gen- 
erating line  is  a  pair  of  straight  lines,  one  of  which  is  the 
generating  line  and  the  other  of  which  meets  every  one  of  the 
generating  lines  mentioned  above. 

Let  the  surface  be  generated  by  the  projective  axial 
pencils  «^i/Sj7j  .  •  •  and  oc^fi^'^^  .... 

Any  plane  that  is  an  element  of  one  of  these  axial 
pencils  intersects  the  surface  in  two  straight  lines,  of 
which  one  line  is  the  base  of  its  pencil,  and  the  other 
line  is  the  line  of  intersection  of  this  plane  with  the  plane 
that  corresponds  to  it  in  the  other  pencil. 

Now  let  a  plane  tt  that  does  not  belong  to  either  axial 
pencil  be  passed  through  a  generating  line  a  which  is  deter- 
mined by  the  planes  a^  and  a^  of  the  axial  pencils. 

This  plane  cuts  the  axial  pencils  in  projective  flat  pencils 
ah^c^  . .  .  and  ah^c^ ....  It  cuts  the  generating  lines  and 
hence  the  surface  generated  in  the  locus  determined  by 
these  flat  pencils.  The  latter  have  a  self-corresponding 
line  a,  and  accordingly  they  are  perspective. 

Consequently  the  locus  determined  by  these  flat  pencils 
consists  of  two  lines ;  namely,  a  and  the  line  in  which  lie 
the  points  of  intersection  of  the  lines  Jj,  h^ ;  c^,  c^;  . .  .. 

The  line  containing  the  points  of  intersection  of  the 
pairs  of  lines  5j,  b^;  c^,  c^;  -  '  •  intersects  every  one  of  the 
generating  lines  determined  by  the  axial  pencils.  For  if 
we  consider  the  line  determined  by  the  corresponding 
planes  /Sj,  ySj,  we  find  that  it  passes  through  the  intersec- 
tion of  the  lines  b^,  b^.  Similarly,  it  may  be  shown  that 
it  meets  the  line  determined  by  any  other  pair  of  corre- 
spondmg  planes  except  a^,  a^.  The  line  and  a,  the  inter- 
section of  a^,  a^,  are  in  the  same  plane  tt,  and  hence  the 
statement  is  established. 


90         PROJECTIVELY  GENEKATED  FIGURES 

5.   Tlie  surface  itself  is  of  the  second  order. 

Let  any  line  p  which  is  not  a  generating  line  intersect 
the  surface  hi  a  point  P.  Pass  a  plane  tt  through  the  line  jt?. 
The  section  of  the  surface  is  a  curve  of  the  second  order, 
and  consequently  the  line  p  cannot  meet  it  in  more  than 
two  points.  Furthermore,  it  cannot  meet  the  surface  in 
points  not  in  this  curve.  Hence  p  cannot  meet  the  sur- 
face in  more  than  two  points.  That  many  lines  actually 
meet  the  surface  in  two  points  follows  from  Case  4.  Hence 
the  surface  is  of  the  second  order. 

96.  A  Range  and  a  Flat  Pencil.  If  a  range  and  a  flat 
pencil  are  projective,  they  may  or  may  not  be  coplanar. 

If  the  range  and  pencil  are  coplanar,  each  point  of  the 
range  taken  with  the  correspondmg  line  of  the  flat  pencil 
determines  the  plane  containmg  both  the  range  and  the  flat 
pencil ;  and  this  plane  is  the  figure  sought. 

If  the  range  and  pencil  are  not  coplanar,  let  the  range 
be  Jj/ijCj  •  •  .  on  the  base  p^,  and  the  flat  pencil  a^b^c^  •  •  • 
on  the  base  ij.  The  plane  determined  by  A^  and  «j  is  the 
same  as  the  plane  determined  by  I^A^  and  a^  Hence  this 
case  yields  the  same  result  as  that  of  two  noncoplanar 
projective  flat  pencils  which  have  a  common  base  ^  (§  87). 

97.  A  Range  and  an  Axial  Pencil.  The  case  of  a  range 
and  an  axial  pencil  may  also  be  considered  briefly. 

A  point  and  a  plane  have  not  been  considered  as  deter- 
mining a  third  element.  If,  however,  the  point  is  in  the 
plane,  they  may  be  said  to  determine  either  the  point  or 
the  plane ;  and  since  all,  two,  one,  or  none  of  the  points 
of  a  range  might  lie  on  the  corresponding  planes  of  an 
axial  pencil  projective  with  the  range,  the  configuration 
sought  in  the  problem  might  be  regarded  as  consisting  of 
pouits  or  planes  as  indicated.    The  case  is  not  important. 


SUMMARY  97 

98.  A  Flat  Pencil  and  an  Axial  Pencil.  The  case  of  a 
flat  pencil  and  an  axial  pencil  demands  but  little  attention. 
There  are  two  subcases,  a  somewhat  trivial  one  in  which 
the  base  of  the  flat  pencil  is  on  the  base  of  the  axial  pencil, 
and  another  subcase  which  has  been  dealt  with  from  a 
different  point  of  view. 

A  little  consideration  shows  that  the  first  subcase  may- 
be regarded  as  yielding  all,  two,  one,  or  none  of  the 
elements  of  the  flat  pencil,  for  these  elements  contain  all 
points  common  to  pairs  of  corresponding  elements  of  the 
flat  pencil  and  the  axial  pencil. 

In  the  second  subcase  the  lines  of  the  flat  pencil  and 
the  corresponding  planes  of  the  axial  pencil  determine  a 
locus  of  points.  But  the  plane  of  the  flat  pencil  cuts  the 
axial  pencil  in  a  second  flat  pencil  projective  with  the  first ; 
and  the  locus  in  question  is  also  determined  by  the  inter- 
sections of  corresponding  lines  of  this  second  flat  pencil 
and  the  given  flat  pencil.  Hence  the  locus  is  the  same  as 
that  described  in  §  86. 

99.  Summary  of  Results.  From  the  discussion  in  this 
chapter  there  have  come  to  notice  the  following  figures, 
exclusive  of  certain  trivial  ones: 

1.  Certain  plane  curves  of  the  second  class. 

2.  Certain  plane  curves  of  the  second  order. 

3.  Certain  conical  surfaces  of  the  second  class. 

4.  Certain  conical  surfaces  of  the  second  order. 

5.  Certain  ruled  surfaces  of  the  second  order  in  space. 

The  study  of  these  curves  and  surfaces  will  be  under- 
taken with  a  view  to  exhibiting  the  symmetry  among  them 
and  to  establishing  their  more  striking  properties,  as  well 
as  with  a  view  to  showing  the  power  of  methods  which 
are  based  upon  the  principles  that  have  been  set  forth. 


98        PKOJECTIVELY  GENERATED  FIGURES 

Exercise  23.   Review 

1.  If  two  points  of  a  circle  are  each  joined  to  four  other 
points  of  the  circle,  the  anharmonic  ratios  of  the  two  pencils  so 
formed  are  equal. 

2.  If  a  point  moving  on  a  circle  is  constantly  joined  to  two 
fixed  points  on  the  circle,  the  flat  pencils  generated  by  the  two 
joining  lines  are  projective. 

3.  If  a  variable  tangent  to  a  circle  meets  two  fixed  tangents, 
the  ranges  traced  by  the  intersections  are  projective. 

s/  4.  A  line  so  moves  as  to  cut  the  sides  BC,  CD  of  a  square 
A  BCD  in  points  A',  Y  such  that  the  angle  XAY  is  constant. 
Find  the  nature  of  the  envelope  of  the  moving  line. 

5.  A  line  so  moves  as  always  to  be  at  a  constant  distance 
from  a  fixed  point.    Find  the  nature  of  the  envelope  of  the  line. 

6.  A  wire  fence  consists  of  a  number  of  horizontal  strands  of 
wire  at  equal  intervals,  crossed  by  a  number  of  vertical  strands 
also  at  equal  intervals  between  each  pair  of  posts.  One  of  two 
posts  is  pushed  into  an  oblique  position.  What  sort  of  surface 
passes  through  all  the  wires  between  the  two  posts  ? 

,^  7.  Two  lighthouses  are  in  a  north-and-south  line.  The  lamps 
revolve  at  the  same  uniform  rate  in  the  same  angular  sense, 
and  each  lamp  throws  two  shafts  of  light  in  opposite  directions. 
If  the  lamps  are  so  adjusted  that  when  one  light  shines  north 
and  south  the  other  shines  northeast  and  southwest,  find  the 
nature  of  the  locus  of  the  spot  illuminated  simultaneously  by 
both  lights.    Has  this  locus  any  infinitely  distant  points  ? 

8.  Consider  Ex.  7  when  thg  lamps  are  so  adjusted  that 
periodically  the  shafts  of  light  coincide. 

9.  Consider  Exs.  7  and  8  when  the  two  lamps  revolve  in 
opposite  senses. 

10.  In  Ex.  7,  if  one  light  rotates  twice  as  quickly  as  the 
other,  do  the  rays  generate  projective  flat  pencils? 


REVIEW  EXERCISES  99 

11.  A  number  of  lamps,  ea«h  of  which  throws  two  shafts  of 
light  in  opposite  directions  and  rotates  at  a  uniform  rate,  are 
to  be  placed  so  that  their  rays  shall  all  continually  converge 
upon  an  object  which  moves  along  a  circle.  If  all  the  lamps 
have  the  same  angular  rate,  specify  a  possible  arrangement. 

12.  In  Ex.  11  specify  an  arrangement  and  adjustment  in 
which  all  the  lamps  do  not  have  the  same  angular  rate. 

13.  Each  of  the  circles  Cj,  c^,  •  •  •,  Cn  is  tangent  to  the  circle 
next  preceding  and  to  the  one  next  following  but  to  no  others 
of  the  set,  and  P^,  P^,  -  •  -,  P^  are  variable  points  on  the  respec- 
tive circles  such  that  each  line  PkPk+i  contains  the  point  of 
contact  of  the  circles  Ck,  c^.+i.  If  0^  is  any  point  on  c^,  and  0„ 
is  any  point  on  c„,  find  the  nature  of  the  locus  of  the  intersec- 
tion of  O^Pj  and  0„P„,  the  figure  being  plane. 

14.  Consider  Ex.  13  with  the  omission  of  the  restriction 
that  no  circle  shall  be  tangent  to  any  other  except  the  one 
next  preceding  and  the  one  next  following. 

15.  Each  of  the  circles  c^,  c^,  •  •  •,  c^  is  tangent  to  the  circle 
next  preceding  and  to  the  one  next  following  but  to  no  others 
of  the  set,  and  t^^,  t,^,  ■  ■  •,  t^  are  variable  tangents  to  the  respec- 
tive circles  such  that  the  point  ^^•^^•+l  is  on  the  common  tangent 
of  c/t,  C/.  +  1.  If  o  is  any  fixed  tangent  to  c^,  and  o„  is  any  fixed 
tangent  to  c„,  find  the  nature  of  the  envelope  of  the  line  join- 
ing the  intersection  of  Oj,  t^  and  that  of  o„,  t^^. 

16.  Each  of  two  circles  in  one  plane  is  divided  into  ten  equal 
arcs.  For  each  circle  the  tangent  at  one  point  of  division  and 
the  secants  through  that  point  and  the  other  points  of  division 
are  drawn.  If  these  two  sets  of  lines  are  produced  indefinitely, 
their  100  points  of  iatersection  lie  in  sets  of  ten  upon  10  curves 
of  the  second  order. 

<  17.  At  the  same  moment  two  trains  leave  a  junction  on 
straight  diverging  lines  and  travel  at  uniform  rates.  If  two 
passengers,  one  on  the  rear  platform  of  each  train,  watch  each 
other,  find  the  envelope  of  their  line  of  sight. 


100       PROJECTIVELY  GENERATED  FIGURES 

18.  Solve  Ex.  17  with  the  modification  that  the  trains  leave 
nearly  but  not  quite  at  the  same  time. 

19.  Each  of  the  right  circular  cones  C^,  C^,  •  •  •,  C^  is  tangent 
along  a  straight  line  to  the  one  following  it.  The  variable  lines 
p  ,  p^  •  •  -ijhi  on  the  respective  cones  are  such  that  each  plane 
PkPkJt\  contains  the  line  of  contact  of  the  cones  C^t,  Cjt+i-  If 
o,,  o„  are  any  fixed  lines  lying  on  Cj,  C„  respectively,  find  the  sur- 
face generated  by  the  intersection  of  the  planes  o^j)^,  o«i>u- 

20.  As  a  train  is  running  along  a  straight  track  at  a  uniform 
rate,  an  automobile  moves,  also  at  a  uniform  rate,  down  a  hill 
along  a  straight  road  which  passes  beneath  the  railroad.  Find 
the  figure  generated  by  the  line  joining  two  fixed  points,  one 
on  the  train  and  the  other  on  the  automobile. 

21.  Initially  a  plane  cuts  two  fixed  intersecting  planes  per- 
pendicularly in  the  lines  o^  and  o^.  As  it  moves  it  cuts  these 
planes  in  the  lines  p^,p^  which  cut  o^  and  o^  at  their  intersection 
and  always  make  equal  angles  with  them.    Find  its  envelope. 

Examine  the  flat  pencils  traced  by  Pj  and  Pg- 

22.  If  A  BCD  is  a  regular  tetrahedron,  and  a  line  so  moves 
as  always  to  intercept  on  AB  and  CD  equal  distances  from 
.1  and  C,  find  the  surface  generated  by  the  line. 

23.  Consider  Ex.  22  if  the  line  continually  divides  AB  and 
CD  proportionally. 

24.  A  sloping  telephone  wire  and  an  electric-light  pole  cast 
upon  the  side  of  a  house  shadows  which  intersect.  If  the  wind 
causes  the  source  of  light  to  swing  in  a  straight  line,  find  the 
path  traced  by  the  intersection  of  the  shadows. 

25.  Consider  Ex.  24  if  the  source  of  light  swings  in  a  circle 
which  intersects  the  wire  and  the  pole. 

26.  Two  fixed  lines  o^  and  o^  intersect  and  pierce  a  plane  w 
at  the  points  O^  and  O^.  Two  planes  tt^  and  tt^  revolve  about  o^ 
and  Oj  respectively  so  that  their  intersections  with  a>  describe 
equal  angles  in  the  same  time.  Find  the  nature  of  the  envelope 
of  the  intersections  of  tTj  and  tt  . 


CHAPTER  IX 

FIGURES  OF  THE  SECOND  ORDER 

100.  Purpose  of  the  Discussion.  The  results  obtained  in 
the  preceding  chapter  may  be  given  a  more  general  as  well 
as  a  more  compact  and  symmetric  form.  It  will  be  noted 
that  these  results  relate  to  three  types  of  figures,  namely, 
figures  in  a  plane ;  figures  in  a  bundle,  or  conical  figures ; 
and  figures  in  space.  These  tliree  types  of  figures  will  be 
considered  separately. 

101.  Plane  Figures.  It  has  already  been  shown  that  the 
figure  obtamed  as  the  envelope  of  the  line  joining  corre- 
sponding points  of  coplanar  projective  ranges  is  a  curve 
of  the  second  class,  and  the  one  obtained  as  the  locus  of 
the  intersections  of  corresponding  lines  of  projective  flat 
pencils  is  a  curve  of  the  second  order.  Whether  all  curves 
of  the  second  class  and  all  curves  of  the  second  order  may 
be  generated  in  this  way,  whether  the  ranges  and  flat  pen- 
cils which  give  rise  to  the  curves  in  question  have  special 
situations  relative  to  those  curves,  and  what  relation,  if  any, 
exists  between  curves  of  the  second  class  and  those  of  the 
second  order,  are  questions  whose  answers  will  exhibit  clearly 
the  importance  and  generality  of  the  results  obtained.  These 
questions  will  now  be  discussed,  but  for  the  sake  of  brevity 
the  treatment  will  be  limited,  the  parts  of  the  argument 
which  are  omitted  being  indicated.  The  student  should 
not  lose  sight  of  the  omission,  and  he  should  later  seek 
to  complete  the  argument  which  answers  the  questions. 

101 


102  FIGUKES  OF  THE  SECOND  ORDER 

102.  Generalization  of  Results.  By  a  line  of  reasoning 
which  will  not  be  given  here  the  following  can  be  proved : 

Every  eurve  of  the  second  class  is  the  envelope  of  the  lines 
joining  corresponding  points  of  two  coplanar  projective  ranges 
ivhose  bases  are  tangent  to  the  curve. 

The  application  of  the  Principle  of  Duality  for  the  plane 
to  this  result  immediately  leads  to  the  further  result : 

Every  curve  of  the  second  order  is  the  locus  of  the  inter- 
sections of  corresponding  lines  of  two  coplanar  projective  flat 
pencils  whose  bases  are  on  the  curve. 

Accordingly  it  is  clear  that  the  developments  in  the 
preceding  chapter  relate  to  all  curves  of  the  second  class 
and  to  all  of  the  second  order. 

Consider  the  second  question  suggested  in  §  101.  We 
have  already  seen  that  the  bases  of  the  projective  ranges 
by  means  of  which  the  curves  of  the  second  class  were 
obtained  are  tangent  to  these  curves  at  certain  of  their 
points.  It  will  be  shown  (§  106)  that  any  two  tangents 
to  a  curve  of  the  second  class  may  be  taken  as  the  bases 
of  projective  ranges  such  that  the  given  curve  is  the 
envelope  of  the  lines  joining  corresponding  points  of  these 
ranges.  Correspondingly,  it  may  be  shown  that  any  two 
pomts  on  a  curve  of  the  second  order  may  be  taken  as 
bases  of  projective  flat  pencils  such  that  the  given  curve 
is  the  locus  of  the  intersections  of  corresponding  lines  of 
these  pencils ;  and  because  to  most  students  the  idea  of  the 
locus  of  points  is  more  familiar  than  that  of  an  envelope, 
the  latter  proposition  will  be  established  first. 

The  proofs  of  these  statements  regarding  curves  of  the 
second  order  depend  upon  an  auxiliary  proposition  which 
will  now  be  stated  and  proved.  Tliis  theorem  will  later 
(§  120)  be  generalized. 


GENERALIZATION  OF  RESULTS  103 

Theorem,  auxiliary  Proposition 

103.  If  ^,  ^,  P^,  ^  are  four  points  on  a  curve  of  the  sec- 
ond order  wliich  is  the  locus  of  the  intersections  of  correspond- 
ing lines  of  the  two  coplanar  projective  fiat  pencils  whose  bases 
are  ^  and  P^,  then  the  pairs  of  lines  I^P^,  P^P^ ;  ^^,  P^P^ ; 
^^,  PqP^  determine  collinear  points. 


Proof.  By  hypothesis  the  points  on  the  curve  in  ques- 
tion determine  projective  flat  pencils  whose  bases  are  I{ 
and  P^.  These  flat  pencils  may  be  denoted  by  -?^(^^-^-^) 
and  P,CP,P,P,P,). 

Cut  these  pencils  by  the  lines  .^^  and  ^^  respectively, 
and  let  the  resulting  ranges  be  Q^^^IQq  and  P^I^R^R^. 

These  ranges  are  not  only  projective  but  also  have  a 
self-corresponding  element  P^.  Hence  they  are  perspective, 
and  the  lines  I^Q^  (or  P^P^^^  P^R^  (or  P^P^-,  and  Q^R^  pass 
through  0,  the  center  of  perspective. 

Therefore  the  points  Q^,  R^  are  collinear  with  the  inter- 
section of  I^P^  and  P^P^.  But  ^g  is  the  intersection  of  P^P^ 
and  i^^,    and  R^  is  the  intersection  of  j^ig  and  ^i^. 

Hence  the  proposition  is  proved. 

By  means  of  the  above  theorem  the  desired  proposition,  known 
as  Steiuer's  theorem,  may  now  be  proved. 


104  FIGURES  OF  THE  SECOND  ORDER 

Theorem.  SiEmER's  Theorem 

104.  Every  curve  of  the  second  order  is  the  locus  of  the 
intersections  of  coplanar  projective  flat  pencils  whose  bases 
are  any  two  points  of  the  curve. 


Proof.  Let  -^,  ^  be  the  bases  of  the  projective  flat 
pencils  which  generate  the  curve,  and  let  -^,  I^,  I^  be  any 
three  fixed  pohits  on  the  curve. 

Let  -^  be  any  point  moving  along  the  curve  and  occu- 
pying successive  positions,  as  j^,  i^',  •  •  •. 

It  will  be  shown  that  the  flat  pencils  generated  by  the 
moving  lines  I^I^  and  im  are  projective. 

For  each  position  of  J^  the  pairs  of  lines  I^,  I^; 
^^,  ^^ ;  I^I^y  P^Il  determine  as  collinear  the  fixed  point 
0  and  the  points  Q^,  R^  (§  103). 

As  II  moves,  Q^  and  R^  move  along  the  fixed  lines  I[Pq 
and  P^Pq  and  trace  ranges  on  them. 

Then  flat  pencil  ^(^^'  •  •  •) 7^  range  R^R^  . .  .(on  ^^) 

=  range  Q^Ql-- "(on  P^P^) 
=  flat  pencil  ^(^^'...). 

Accordingly  the  curve  is  the  locus  of  the  intersections 
of  corresponding  lines  of  the  projective  flat  pencils  whose 
bases  are  I^  and  ^,  any  two  points  of  the  curve. 


STEINER'S  THEOREM  105 

Theorem.  Auxiliary  proposition 

105.  If  <2»  '3,  ^4,  ^6  ^^^  /•'^^  tangents  to  a  curve  of  the 
second  class  which  is  the  envelope  of  the  lines  joining  cor- 
responding points  of  two  coplanar  projective  ranges  whose 
bases  are  ^  and  t^,  then  the  pairs  of  points  t^t^,  t^t^ ;  t^t^,  t^t^ ; 
^3^4,  fg^j  determine  concurrent  lines. 

This  theorem  is  the  dual  of  the  theorem  of  §  103  and  leads  to  the 
dual  of  Steiner's  theorem.  It  will  later  (§  121)  be  generalized  into 
a  highly  important  proposition.    The  proof  is  left  for  the  student 

Theorem.  Dual  of  Steiner's  Theorem 

106.  Every  curve  of  the  second  class  is  the  envelope  of  the 
lines  joining  corresponding  points  of  the  coplanar  projective 
ranges  whose  bases  are  any  two  tangents  to  the  curve. 

It  is  particularly  important  that  by  means  of  the  Principle  of 
Duality,  or  otherwise,  the  student  should  follow  out  in  detail  the 
proof  of  this  proposition,  as  well  as  the  proof  of  the  proposition  which 
immediately  precedes  it  (§  105).  There  is  not  much  difficulty  in 
obtaining  the  steps  of  the  proof  as  duals  of  the  corresponding  steps 
of  the  proof  in  §  104,  but  the  figure  and  the  verification  of  the  various 
steps  of  the  argument  in  connection  with  this  figure  require  close 
attention  on  the  part  of  the  student. 

107.  Relations  between  Curves  of  the  Second  Order  and 
Curves  of  the  Second  Class.  The  two  sorts  of  plane  curves 
which  have  been  obtained  can  now  be  shown  to  be  iden- 
tical. In  other  words,  it  can  be  shown  that  every  curve 
of  the  second  order  is  of  the  second  class,  and  conversely. 
Only  one  of  these  proofs  will  be  given,  since  the  other  can 
be  derived  from  it  by  means  of  the  Principle  of  Duality. 
In  this  proof  use  will  be  made  of  a  limitmg  case  of  the  prop- 
osition in  §103,  in  which  two  of  the  four  arbitrarily  chosen 
points  on  the  curve  have  moved  up  to  coincidence  with 
the  bases  of  the  pencils,  and  this  will  first  be  established. 


106  FIGURES  OF  THE  SECOND  ORDER 

Theorem.  Inscribed  Quadrangle 

108.  If  I^y  Jl  are  two  points  on  a  curve  of  the  second 
order  which  is  generated  hy  two  coplanar  projective  fiat  pen- 
cils whose  bases  are  i^  and  ^,  the  tangents  at  J^  and  ^,  the 
tangents  at  i^  aiid  i^,  and  the  pairs  of  opposite  sides  I^I^, 
im;  J^I^,  -^i^  of  the  inscribed  quadrangle  imi^I^  inter- 
sect in  collinear  points. 


Proof.  In  the  figure  of  §  103  let  ^  and  ^  move  along 
the  curve  so  as  to  approach  ^  and  i^  as  limiting  points. 

Then  I^  and  I^  approach  the  tangents  to  the  curve  at 
J^  and  ^  respectively,  and  0  approaches  the  intersection  of 
the  tangents  at  ^  and  ^  as  a  limiting  point. 

The  hexagon  I^I^I^P^P^Il  approaches  the  quadrangle 
im^Il,  together  with  the  tangents  at  ^  and  I^. 

During  the  motion  of  the  points  and  lines  the  intersec- 
tions of  the  pairs  of  lines  -^^,  ^^;  ^^,  ^^;  ^^,  i^i^ 
remain  colUnear,  and  the  limiting  positions  which  they 
approach  are  collinear. 

Since  I{,  P^  are  not  special  points  on  the  curve  (§  104), 
it  follows  that  the  tangents  at  J3,  Pq  meet  on  the  same  line. 

Then  the  theorem  of  §  103  takes  the  form  of  this  theorem. 

This  proposition  will  be  used  as  auxiliary  to  the  proof  of  the 
identity  of  curves  of  the  second  order  with  those  of  the  second  class. 


STEINER'S  THEOREM  107 

Exercise  24.    Steiner's  Theorem  and  Related  Theorems 

1.  In  the  theorem  of  §  103,  if  P^,  P^,  P^,  P^  are  fixed  points, 
and  if  P„  and  P.  so  "move  that  the  intersection  R  otPP.PP 

o  D  6  2     8'       5      6 

moves  on  a  fixed  line  through  the  intersection  of  P^P^,  P^P  , 
then  the  intersection  Q^  of  P^P^,  P^P^  moves  on  the  same  line, 
and  Qg,  R^  trace  on  this  line  two  superposed  projective  ranges. 

2.  In  Ex.  1  find  the  self-corresponding  points  of  the  super- 
posed projective  ranges. 

3.  Prove  the  proposition  which  is  related  to  the  theorem  of 
§  108  as  Ex.  1  is  related  to  the  theorem  of  §  103. 

4.  Solve  the  problem  which  is  related  to  Ex.  3  as  Ex.  2  is 
related  to  Ex.  1. 

5.  If  fg,  fg  are  two  tangents  to  a  curve  of  the  second  class 
which  is  the  envelope  of  the  lines  joining  corresponding 
points  of  two  coplanar  projective  ranges  whose  bases  are  t^,  t^, 
the  points  of  contact  of  t^,  t^,  the  points  of  contact  of  t^,  t^, 
and  the  pairs  of  opposite  vertices  t^t^,  t^t^  ;  t^t^,  t^t^  of  the 
circumscribed  quadrilateral  t/^t^t^  determine  concurrent  lines. 

6.  A  variable  hexagon  ^i^2^3^4^5^6  ii^scribed  in  a  curve 
of  the  second  order  so  moves  that  P,P„,  P.P.:  P„P„,  P^P. 
always  intersect  in  fixed  points  O  and  R  respectively.  Find 
the  locus  of  the  intersection  of  P^P^,  P^P^ 

7.  Prove  the  duals  of  Exs.  1,  3,  and  6  for  the  plane. 

8.  Solve  the  dual  of  Ex.  2  for  the  plane. 

9.  By  an  argument  independent  of  that  given  in  this  chap- 
ter prove  Steiner's  theorem  for  the  special  case  of  the  circle. 

10.  By  an  argument  independent  of  that  referred  to  in  §  106 
prove  the  theorem  stated  there  for  the  special  case  of  the  circle. 

11.  Pj,  Pg,  Pg,  P^,  P5  are  five  fixed  coplanar  points  no  three 
of  which  are  in  a  straight  line;  find  the  locus  of  a  point 
P  which  so  moves  that  the  intersections  of  the  pairs  of  lines 
P^P^,  P^P^ ;  P^P^,  P^P ;  P^P^,  PP^  are  constantly  collinear. 


108  FIGURES  OF  THE  SECOND  ORDER 

Theorem.  Identity  op  Curves 

109.  Every  plane    curve    of  the   second  order  is  of  the 
second  class,  and  conversely. 


Proof.  When  a  curve  of  the  second  order  is  given,  the 
pencils  of  lines  drawn  from  two  of  its  points  to  all  of 
its  points  are  projective.  From  any  three  pairs  of  corre- 
sponding lines  all  additional  pairs  can  be  obtained  by  the 
method  of  §  39.  In  particular,  since  the  tangent  at  either 
of  the  two  points  corresponds  to  the  line  joining  the  two 
points,  it  can  be  drawn  by  the  same  method;  and  there- 
fore, when  the  whole  curve  is  given,  the  tangents  at  as 
many  points  as  may  be  desired  can  be  drawn. 

Let  a  curve  of  the  second  order  be  given,  and  select 
on  it  any  three  points  ^,  ^,  ^.  Draw  J^,  ^^,  i^^  and 
the  tangents  MI^N,  NI^L,  and  LI^M.  The  projectivity  is 
determined  by  the  triads  I^M,  ij^,  J^  and  ^i^,  j^i,  I^I^. 

Consider  the  curve  as  the  locus  of  a  moving  point  P. 
It  will  be  shown  that  as  P  moves  along  the  curve  the 
tangent  at  P  so  moves  as  to  meet  the  tangents  at  i^  and  ^ 
in  corresponding  points  X  and  Y  of  two  projective  ranges. 


IDENTITY  OF  CURVES  109 

Let  0,  K  be  the  intersections  of  the  sides  -^j^,  I^P  and 
^P,  I^  of  the  quadrangle  PI{I^I^  inscribed  in  the  given 
curve.  For  this  inscribed  quadrangle,  M  is  the  intersection 
of  tangents  at  opposite  vertices  and  0,  K  are  intersections 
of  pairs  of  opposite  sides,  and  hence  it  follows  from  §  108 
that  the  points  M,  0,  K  are  collinear.  By  means  of  a  like 
reasonmg  the  points  0,  K,  Y  are  proven  collinear.  Hence 
the  points  M,  0,   Y  are  collinear. 

Similarly,  the  pomts  L,  0,  X  may  be  proved  collinear. 

Since  ^,  -^,  ^  are  fixed  points,  the  tangents  at  these 
points  are  also  fixed.  As  the  point  P  moves,  so  do  the 
lines  P^P,  P^P,  P^P,  XY,  l^O,  LO,  MO,  and  the  points 
0,  X,  Y.  The  Imes  LO,  MO  generate  perspective  flat  pen- 
cils with  bases  at  L  and  M,  and  the  points  X  and  Y  trace 
on  the  lines  MI^N  and  LI^N  ranges  perspective  with  these 
pencils  and  hence  projective  with  each  other.  Therefore, 
as  P  moves,  the  tangent  at  P  so  moves  as  always  to  meet 
the  tangents  at  J^  and  ^  in  correspondmg  points  of  two 
projective  ranges. 

The  given  curve  is  the  envelope  of  the  tangent  at  P 
and,  by  §  104,  is  of  the  second  class. 

The  converse  of  this  proposition  being  also  its  dual  for 
the  plane,  its  proof  is  also  the  dual  of  the  above  argu- 
ment.   The  theorem  is  therefore  established. 

110.  Conic.  A  plane  curve  which  is  of  the  second  order 
and  second  class  can  be  shown  to  be  a  curve  ordinarily 
called  a  conic  section  or  a  conic. 

The  proof  will  not  be  given,  but,  independently  of  its  other  uses, 
the  word  conic  will  be  employed  to  designate  any  curve  of  the 
second  order.  The  use  of  properties  of  conies  not  deduced  from  the 
definition  here  given  will  be  avoided. 

A  conic  section,  or  conic,  is  a  curve  of  the  second  order 
and  second  class. 


110  FIGURES  OF  THE  SECOND  ORDER 

Theorem,  order  and  class  of  Surfaces  (m  the  Bundle) 

111.  Every  surface  (in  the  bundle)  that  is  of  the  second 
order  is  of  the  second  class,  and  conversely. 

Proof.  A  comparison  of  the  problems  whose  discussion 
led  to  §§84  and  90  shows  that  these  theorems  deal  with 
figures  and  state  results  that,  for  threefold  space,  are  dual. 
Similarly,  the  theorems  of  §§86  and  87  are  dual.  It  was 
indicated,  but  not  proved,  that  §  84  is  true  for  all  plane 
curves  of  the  second  class  and  that  §  87  is  true  for  all 
plane  curves  of  the  second  order.  Correspondingly,  §§90 
and  87  can  be  shown  to  apply  to  all  surfaces  (in  the 
bundle)  that  are  of  the  second  class  and  all  surfaces  (in 
the  bundle)  that  are  of  the  second  order.  Moreover,  the  iden- 
tity of  the  curves  of  the  second  order  with  curves  of  the 
second  class  having  been  established  in  §  109,  the  identity  of 
these  surfaces  of  the  second  order  with  those  of  the  second 
class  may  be  established  by  reasoning  dual  to  that  of 
§  109.  This  involves  the  derivation  of  an  auxiliary  theorem 
dual  for  space  to  that  of  §  103  and  one  that  is  dual  to  the 
limiting  case  of  the  latter  as  worked  out  in  §  108.  The 
student  will  find  that  the  principal  difficulty  is  connected 
with  the  drawing  of  appropriate  figures  for  these  cases. 

In  this  way  the  classes  of  figures  numbered  3  and  4  in  §  99  are 
shown  to  be  identical. 

112.  Quadric.  Since  a  surface  (in  the  bundle)  of  the 
second  order  and  second  class  may  be  thought  of  as  gener- 
ated by  the  motion  of  a  line  that  always  passes  through 
a  fixed  point,  it  is  said  to  be  a  conic  surface  or  a  cone ; 
and  on  account  of  its  order  and  class  it  is  called  a 
quadric  conic  surface. 

A  quadric  conic  surface  or  quadric  cone  is  a  surface  (in 
the  bundle)  of  the  second  order  and  second  class. 


ORDER  AND  CLASS  OF  SURFACES  111 

Theorem.  Skew  Ruled  Surfaces 

113.  Every  ruled  surface  (not  in  the  plane  or  in  the  bundle') 
of  the  second  order  is  of  the  second  cla^s^  and  conversely. 

Proof.  In  §§85  and  95  certain  ruled  surfaces  of  the 
second  order  that  exist  in  threefold  space  but  not  in  the 
plane  or  in  the  bundle  were  found  to  be  generated  by  means 
of  both  projective  ranges  and  projective  axial  pencils. 
It  can  further  be  shown  that  all  ruled  surfaces  of  the 
second  order  can  be  so  generated.  Moreover,  a  complete 
discussion  of  these  surfaces  from  the  point  of  view  of 
both  methods  of  generation  would  have  led  to  results 
similar  to  those  obtained  for  the  conies  and  for  the  quadric 
cones.  Among  other  things  it  would  have  appeared  that 
the  planes  which  pass  through  a  point  P  not  on  the  sur- 
face, and  are  tangent  to  the  surface,  would  generate  a 
quadric  cone,  and  that  of  these  planes  not  more  than  two, 
but  in  some  cases  two,  would  pass  through  a  line  that 
contains  F.    Hence  these  surfaces  are  of  the  second  class. 

The  converse  may  also  be  proved. 

The  discussion  indicates  that  the  class  of  figures  numbered  5  in 
§  99  is  self-dual  in  threefold  space. 

114.  Summary.  We  have  now  shown  that  the  config- 
urations whose  generating  elements  are  determined  by 
corresponding  elements  of  two  projective  one-dimensional 
prime  forms  are  as  follows: 

1.  All  plane  curves  of  the  second  order  and  second  class 
(conies). 

2.  All  conic  surfaces  of  the  second  order  and  second  cla^s 
(quadric  cones'). 

3.  All  ruled  mrfaces  of  the  second  order  and  second  class 
not  in  the  plane  or  in  the  bundle. 


112  FIGURES  OF  THE  SECOND  ORDER 

Exercise  25.   Review 

1.  State  and  prove  the  dual  for  space  of  §  103. 

2.  State  and  prove  the  dual  for  the  bundle  of  Ex.  1. 

3.  Compare  the  dual  for  space  of  the  theorem  in  §  105 
with  the  result  of  Ex.  2. 

4.  Give  the  dual  for  the  plane  of  the  statement  and  proof 
of  Ex.  11,  page  107. 

5.  Give  the  dual  for  space  of  the  statement  and  proof 
of  the  theorem  in  §  108. 

6.  Compare  the  space  dual  of  the  result  of  Ex.  5,  page  107, 
with  the  dual  for  the  bundle  of  the  result  of  Ex.  5,  above. 

7.  Derive  the  dual  for  space  of  Ex.  1,  page  107. 

8.  Establish  three  projectivities  between  flat  pencils  which 
shall  lead  to  the  generation  of  conies  having  respectively  two, 
one,  and  no  points  of  intersection  with  any  given  straight  line. 

In  this  connection  consider  §  68. 

9.  Consider  Ex.  8  for  the  case  in  which  the  given  straight 
line  is  the  line  at  infinity  of  a  given  plane. 

Tlie  student  will  observe  that  the  solution  of  this  problem  establishes 
the  existence  of  conies  with  two,  one,  and  no  points  at  infinity. 

10.  Establish  three  projectivities  between  ranges  which 
shall  lead  to  the  development  of  conies  having  respectively 
two,  one,  and  no  tangents  whose  points  of  contact  are  on 
any  straight  line. 

11.  Consider  Ex.  10  for  the  case  in  which  the  given  straight 
line  is  the  line  at  infinity  of  a  given  plane. 

12.  Prove  the  dual  for  space  of  Ex.  6  on  page  107. 

13.  Solve  the  dual  for  the  plane  of  Ex.  8. 

14.  Consider  Ex.  13  for  the  case  in  which  the  given  point 
is  at  infinity  in  a  given  direction. 

15.  Solve  the  dual  for  space  of  Ex.  8. 


REVIEW  EXERCISES  113 

16.  Five  concurrent  lines,  no  three  of  which  are  in  any  one 
plane,  all  lie  on  one  conic  surface  of  the  second  order. 

17.  Prove  the  proposition  regarding  five  parallel  lines  which 
corresponds  to  Ex.  16. 

18.  Establish  three  projectivities  between  axial  pencils 
which  shall  lead  to  the  generation  of  conic  surfaces  having 
their  vertices  at  infinity  and  having  as  right  sections  curves 
with  two,  one,  and  no  points  at  infinity  respectively. 

19.  Establish  between  two  given  ranges  which  are  not  in 
the  same  plane  a  projectivity  such  that  if  the  surface  gener- 
ated is  cut  by  any  given  plane  in  the  finite  part  of  space,  the 
section  shall  be  two  straight  lines. 

20.  In  a  bundle  tTj,  tt^,  tt^,  tt^,  tt^  are  five  fixed  planes,  no 
three  of  which  are  coaxial.  Find  the  envelope  of  a  plane  ir 
which  moves  so  that  the  planes  determined  by  the  intersec- 
tions   of   TT^,  TT^  and  TT^,  TTg,  of  TT^,  TTg  and  TTg,  TT,  and  of    TTg,  TT^ 

and  TT,  TT^  are  constantly  coaxial. 

21.  Consider  Ex.  19  for  the  case  in  which  the  given  plane 
is  at  infinity. 

22.  Given  a  plane  and  two  projective  axial  pencils  which 
have  no  common  element,  establish  between  other  pencils  a 
projectivity  which  shall  lead  to  the  generation  of  a  surface 
that  shall  be  the  projector  from  a  given  center  of  the  intersec- 
tion of  the  given  plane  and  the  surface  generated  by  the  given 
axial  pencils. 

23.  Derive  the  dual  of  Ex.  22  for  space. 

24.  Given  in  a  plane  three  nonconcurrent  bases  p^,  p^,  p^ 
passing  through  the  points  A^,A^,A^  respectively,  specify  three 
projectivities  which  connect  ranges  on  the  bases  p^,p^'i  P^tPs'^ 
p^,p^  respectively,  and  which  are  such  that  the  three  conies 
that  they  determine  shall  coincide  and  also  be  tangent  to  the 
three  lines  p^,  p,^,  p^  at  A^,  A^,  A^  respectively. 

25.  Examine  Ex.  24  for  the  case  in  which  the  three  points 
A^,  A^,  A^  are  collinear. 


114  FIGURES  OF  THE  SECOND  ORDER 

26.  In  Ex.  24  select  such  lines  p^,  p^,  p^  and  such  points 
A^,  A.^,  A^  that  the  conic  generated  shall  be  a  circle. 

27.  Given  in  a  bundle  three  non-coaxial  planes  tTj,  tt^,  tt, 
passing  through  the  lines  a^,  a^,  Og  respectively,  specify  three 
projectivities  which  connect  flat  pencils  in  the  planes  tTj,  tt^; 
TTjj,  TTg;  TTg,  TT^  respectively,  and  which  are  such  that  the  three 
conic  surfaces  they  determine  shall  coincide  and  shall  be 
tangent  to  the  planes  tt^,  tt^,  tTj  along  the  lines  a^,  a.^,  a^. 

28.  Solve  the  dual  of  Ex.  27  for  the  bundle. 

29.  Consider  Ex.  19  for  the  case  in  which  the  section  by  the 
plane  at  infinity  is  to  be  two  straight  lines. 

30.  Establish  such  a  projectivity  between  two  given  axial 
pencils,  not  in  the  same  bundle,  that  if  the  surface  generated 
is  cut  by  any  given  plane  in  the  finite  part  of  space,  the  section 
shall  be  circular. 

31.  Find  two  axial  pencils,  not  in  the  same  bundle,  between 
which  such  a  projectivity  may  be  established  that  the  corre- 
sponding surface  generated  shall  be  cut  by  a  given  plane  in  a 
given  circle  of  that  plane. 

32.  Given  two  projective  axial  pencils,  not  in  a  bundle,  pass 
a  plane  which  shall  cut  the  surface  generated  by  them  in  two 
straight  lines. 

33.  Given  three  bases  p^,  p^,  p^  in  space,  no  two  of  which 
intersect,  specify  three  projectivities  between  ranges  on  the 
bases  p^,  p^\  p.^,  p^\  p^,  j)^  respectively,  such  that  the  three  skew 
ruled  quadric  surfaces  determined  by  them  shall  coincide. 

34.  Solve  the  dual  for  space  of  Ex.  33. 

35.  Develop  completely  the  proof  of  the  theorem  corre- 
sponding to  the  theorem  of  §  109  for  the  case  of  figures  of 
the  second  order  in  the  bundle. 

36.  For  the  case  of  figures  of  the  second  order  in  three- 
fold space  describe  accurately  the  figure  for  the  theorem 
corresponding  to  that  of  §  109,  and  outline  the  proof. 


CHAPTER  X 
CONICS 

115.  Detennination  of  Conies  by  Certain  Conditions.  Some 
of  the  more  important  properties  of  the  curves  and  sur- 
faces to  which  attention  has  been  drawn  in  the  preceding 
chapters  will  now  be  deduced.  The  conies  will  be  dealt 
with  much  more  fully  than  the  other  figures  because  of 
their  more  frequent  application  and  also  because,  after 
their  properties  have  been  set  forth,  the  corresponding 
properties  of  the  quadric  cone  may  be  obtained  by  means 
of  the  Principle  of  Duality. 

Chapters  X-XII  are  devoted  to  the  conies. 

In  Chapter  XIII  there  is  given  a  discussion  of  quadric 
cones,  this  being  confined  to  a  few  topics  in  addition  to  those 
suggested  by  the  developments  obtamed  for  the  conies. 
Notwithstanding  this  limitation,  students  should  give  due 
attention  to  these  properties  of  quadric  cones. 

In  Chapter  XIV  will  be  found  a  brief  introduction  to 
the  study  of  the  properties  of  skew  quadric  ruled  surfaces. 
A  thorough  study  of  these  figures  may  well  be  deferred 
until  the  student  has  an  opportunity  to  approach  the  sub- 
ject from  the  point  of  view  of  analytic  geometry  also,  when 
a  comparison  of  the  analytic  and  synthetic  treatments  will 
heighten  the  interest. 

The  first  body  of  facts  to  be  established  relates  to  sets 
of  data  which  completely  determine  conies.  It  constitutes 
the  important  theorem  stated  in  §  116,  which  consists  of 
six  simple  propositions. 

115 


116  CONICS 

Theorem.  A  Conic  Determined 

116.  In  a  plane  there  is  one  and  only  one  conic  which  has 
one  of  the  following  properties : 

1.  It  passes  through  five  given  points,  no  four  of  which  are 
collinear. 

2.  It  passes  through  four  given  points,  no  three  of  which 
are  collitiear,  and  at  any  one  of  these  points  is  tangent  to  a 
given  line  which  passes  through  this  point  hut  not  through 
any  other  of  the  given  points. 

3.  It  passes  through  three  given  points,  not  collinear,  and 
at  each  of  two  of  these  points  is  tangent  to  a  given  line  which 
passes  through  this  point  but  not  through  any  other  of  the 
three  given  points. 

4.  It  passes  through  tivo  given  points,  and  at  each  of  these 
is  tangent  to  a  given  line  which  parses  through  that  point  hut 
not  through  the  other  given  point,  and  in  addition  is  tangent 
to  a  third  given  line  which  is  not  concurrent  with  the  other 
two  given  lines. 

5.  It  parses  through  a  given  point  and  is  tangent  at  that 
point  to  a  given  line  through  the  point,  and  is  tangent  to 
each  of  three  other  given  lines  so  situated  that  of  four  given 
lines  no  three  are  concurrent. 

6.  It  is  tangent  to  five  given  lines,  no  four  of  which  are 
concurrent. 

Without  doubt  the  student  will  generally  use  in  his  work  an 
abridgment  of  this  statement.  The  longer  statement  given  above 
may  be  regarded  as  an  interpretation  of  the  shorter  one  in  §  117, 
making  clear  the  meaning  of  the  determination  of  a  figure  by  means 
of  certain  data.  The  student  will  find  that  very  frequently  in  geom- 
etry this  abridged  form  of  statement  is  used  in  the  sense  expressed 
more  fully  by  the  other  one.  Occasional  expansions  of  shorter 
statements  into  the  corresponding  longer  ones  are  well  worth  the 
attention  of  the  student. 


A  CONIC  DETERMINED  117 

Theorem.  Alternative  Statement  of  §  116 

117.  A  conic  is  determined  by  any  one  of  the  following 
sets  of  elements  that  are  associated  with  it: 

1.  Five  of  its  points. 

2.  Four  of  its  points  and  the  tangent  at  one  of  these  points. 

3.  TJiree  of  its  points  and  the  tangents  at  two  of  these  points. 

4.  TJiree  of  its  tangents  and  the  points  of  contact  on  two  of 
these  tangents. 

5.  Four  of  its  tangents  and  the  point  of  contact  on  one 
of  these  tangents. 

6.  Five  of  its  tangents. 

Proof.  We  shall  deal  with  the  cases  in  the  above  order. 
1.  A  conic  is  determined  by  five  of  its  points. 


Let  i^,  i^,  ^,  i^,  i^  be  five  points  of  a  plane,  no  four  of 
them  being  coUinear.    Join  jP  and  i^  to  i^,  ^,  P^. 

The  triads  of  lines  P^P^,  P^P^,  P^P^  and  ^i^,  ^i^,  P^P^ 
determine  a  projectivity  between  the  flat  pencils  whose 
bases  are  P^,  ^,  and  hence  they  determine  a  conic  through 
the  five  points.  Any  conic  through  these  points  could  be 
generated  from  the  projectivity  determined  by  the  same 
triads  (§  104)  and  would  be  the  same  as  the  one  mentioned. 

If  three  of  the  five  points  are  situated  on  a  line  I,  the  other  two 
points  should  be  taken  as  the  bases  of  the  pencils.  In  this  case 
the  conic  consists  of  the  line  I  and  the  line  through  the  bases. 


118  CONICS 

2.  A  conic  is  determined  hy  four  of  its  points  and  the 
tangent  to  it  at  one  of  these  points. 


Let  ^,  J^,  I^,  II  be  four  points  of  a  plane,  no  three  of 
them  being  collinear,  and  in  the  plane  let  t^  be  any  line 
which  passes  through  i^  but  which  does  not  pass  through 
any  other  of  the  four  given  points. 

Draw  from  7J  the  Imes  I^,  I^I^,  and  draw  from  J^  the 
lines  P^P,,  P^P^,  P^P,. 

Consider  ^  and  ^  as  bases  of  flat  pencils.  The  lines  t^, 
J^P^ ;  ^^,  I^P^ ;  J^^,  ^^  being  taken  as  corresponding,  one 
and  only  one  projectivity  is  thereby  established  between 
the  flat  pencils.  This  projectivity  determines  one  conic 
passing  through  the  points  ij,  ^,  ^,  P^  and  having  the 
line  t^  as  its  tangent  at  I{. 

No  other  conic  can  fulfill  these  conditions,  since  in  that 
case  the  conic  would  also  be  generated  from  the  projectivity 
just  mentioned,  and  hence  this  conic  would  coincide  with 
the  first  one. 

Therefore  the  second  statement  is  proved. 

If  three  of  the  four  points  other  than  the  one  at  which  the  tangent 
is  given  are  on  a  line  I,  the  conic  consists  of  the  given  tangent  and 
the  line  I.  If  Pj  and  two  only  of  the  other  points  are  collinear,  no 
conic  is  determined. 


A  CONIC  DETERMINED  •       119 

3.  A  conic  is  determined  hy  three  of  its  points  and  the 
tangents  to  it  at  tivo  of  these  points. 


tz 


Let  ^,  ^,  P^  be  three  noncollinear  points  of  a  plane, 
and  in  the  plane  let  t^  and  t^  be  lines  which  pass  through 
j^  and  -^  respectively  but  through  no  other  of  the  three 
given  points.    Draw  the  lines  i^^,  ^^,  and  I^P^. 

The  triads  t^,  I^,  I^P^  and  -^^,  t^,  I^P^  determine  a  pro- 
jectivity  between  the  pencils  whose  bases  are  I[  and  ^, 
and  this  projectivity  determines  a  conic  passing  through 
^,  ^,  P^  and  tangent  at  ^  and  ^  to  t^  and  t^  respectively. 

As  in  the  other  cases,  it  may  be  shown  that  there  is 
only  one  such  conic.    Hence  the  third  statement  is  proved. 

If  the  three  points  are  on  a  line  I  and  if  one  of  the  given  tangents 
is  I,  the  conic  consists  of  I  and  the  other  tangent.  If  neither  of  the 
tangents  coincides  with  I  or  both  tangents  coincide  with  I,  the  conic 
may  be  thought  of  as  the  line  I  taken  twice. 

4.  A  conic  is  determined  hy  three  of  its  tangents  and  the 
points  of  contact  of  two  of  these  tangents. 

Since  Nos.  3  and  4  are  dual  in  the  plane,  the  proof  of 
No.  4  follows  at  once. 

5.  A  conic  is  determined  by  four  of  its  tangents  and  the 
point  of  contact  of  one  of  these  tangents. 

Since  Nos.  2  and  5  are  dual  in  the  plane,  the  proof  of 
No.  5  follows  at  once. 

6.  A  conic  is  determined  hy  five  of  its  tangents. 

Since  Nos.  1  and  6  are  dual  in  the  plane,  the  proof  of 
No.  6  follows  at  once. 


120      .  CONICS 

118.  Construction  of  Conies.  The  statements  of  §  117 
establish  the  existence  of  conies  that  fulfill  certain  condi- 
tions, and  suggest  but  do  not  solve  the  problem  of  con- 
structing the  conies  under  these  various  conditions.  The 
solution  of  this  problem  can  be  based  upon  the  notion 
of  projectivity  involved  in  §  117,  but  it  can  also  be  based 
upon  two  very  celebrated  theorems  which  will  be  considered 
on  page  121.  After  these  theorems  have  been  proved,  the 
problems  of  the  construction  of  conies  will  be  treated  from 
both  points  of  view. 

Before  considering  these  two  theorems,  however,  it  will 
be  found  necessary  to  make  some  extension  of  the  common 
notion  of  a  hexagon  with  which  the  student  is  familiar  from 
elementary  geometry. 

119.  Hexagon.  If  any  six  coplanar  points  are  taken  in 
a  given  order,  the  figure  formed  by  the  lines  through  all 
pairs  of  successive  points,  as  well  as  through  the  first  and 
last  points,  is  called  a  hexagon. 


As  in  the  ordinary  case  of  the  hexagon,  the  first  and 

fourth,  the  second  and  fifth,  and  the  third  and  sixth  sides 

are  called  opposite  sides. 

Thus,  in  the  above  figures  the  pairs  of  opposite  sides  are  P-yP^, 
p  p  .  p  p     p  p  .  p  p     p  p 

In  each  of  the  above  figures  the  diagonals  from  P^  are  PiP^, 
P,P^,  and  P^P,. 

A  similar  generalization  applies  to  each  of  the  other  polygons.  It 
thus  ajjpears  that  opix)site  sides  of  a  quadrilateral  may  intersect  and 
that  a  diagonal  may  lie  wholly  outside  a  polygon. 


THEOREMS  OF  PASCAL  AND  BEIANCHON   121 
Theorem.  Pascal's  Theorem 

120.  If  a   hexagon   is   inscribed   in   a   conic,    the   three 
intersections  of  the  three  pairs  of  opposite  sides  are  collinear. 

f3 


Proof.    This  is  the  theorem  of  §  103  with  the  restriction 
upon  i^,  ^  removed  by  Steiner's  theorem  (§  104). 

Unless  a  cross  hexagon  is  taken,  the  figure  is  usually  very  large. 
The  proposition  is  due  to  Blaise  Pascal  (1623-1662). 


Theorem.  Brianchon's  Theorem 

121.  If  a  hexagon  is  circumscribed  about  a  conic,  the  three 
lines  joining  the  three  pairs  of  opposite  vertices  are  concurrent. 


Proof.    This  is  simply  a  generalization  of  §  105. 

The  student  should  write  out  the  proof  of  this  theorem. 

The  proposition  is  due  to  Charles  Julien  Brianchou  (1785-1864). 


122  CONICS 

122.  Pascal  Line.  The  line  containing  the  points  of  inter- 
section of  the  three  pairs  of  opposite  sides  of  a  hexagon 
in  a  conic  is  called  the  Pascal  line  of  the  hexagon. 

123.  Brianchon  Point.  The  point  of  concurrence  of  the 
three  lines  joining  the  opposite  vertices  of  a  hexagon  about 
a  conic  is  called  the  Brianchon  point  of  the  hexagon. 

124.  Converses  of  the  Theorems  of  Pascal  and  Brianchon. 
The  converses  of  the  theorems  of  Pascal  and  Brianchon 
can  be  established  as  in  the  exercise  below,  and  each  of 
them  may  then  be  given  a  different  interpretation.  Thus, 
if  six  coplanar  points  are  chosen  and  joined  to  form  a  hexa- 
gon, a  conic  passes  through  any  five  of  them.  Does  it  pass 
through  the  sixth  point  ?  It  does  if  and  only  if  the  three 
points  of  intersection  of  the  pairs  of  opposite  sides  are 
collinear.  Hence  Pascal's  theorem  and  its  converse  imply 
the  necessary  and  sufficient  conditions  for  the  passing  of 
a  conic  through  six  given  coplanar  points.  Brian chon's 
theorem  can  be  interpreted  in  a  corresponding  fashion. 

Exercise  26.    Theorems  of  Pascal  and  Brianchon 

1.  State  and  prove  the  converse  of  Pascal's  theorem. 

2.  If  two  pairs  of  opposite  sides  of  a  hexagon  inscribed  in 
a  conic  are  parallel,  the  other  two  opposite  sides  are  parallel. 

3.  A  hexagon  is  to  be  inscribed  in  a  conic  in  such  a  way  that 

a  given  line  shall  be  its  Pascal  line.  Determine  the  maximum 
number  of  sides  of  the  hexagon  that  may  be  given,  and  solve 
the  problem. 

4.  Solve  Ex.  3  for  the  ease  when  the  given  line  is  at  infinity. 

5.  State  and  prove  the  converse  of  Brianchon's  theorem, 

6.  Circumscribe  a  hexagon  about  a  given  conic  in  such  a  way 
that  a  given  point  shall  be  its  Brianchon  point,  as  many  of  the 
vertices  of  the  hexagon  as  possible  being  given  in  advance. 


THEOREMS  OF  PASCAL  AND  BRIANCHON   123 

125.  Limiting  Cases  of  the  Theorems  of  Pascal  and  Brian- 
chon.  There  are  several  limiting  cases  of  the  theorems 
of  Pascal  and  Brianchon  which  have  useful  applications 
and  which  require  mention  at  this  point.  They  arise  out  of 
approach  to  coincidence  of  vertices  of  an  inscribed  hexagon 
of  a  conic  and  also  of  sides  of  a  circumscribed  hexagon. 

Let  im^P^P^P^  be  a  hexagon  mscribed  in  a  conic.  If  ^ 
approaches  I{  along  the  conic,  the  line  I^  approaches  the 
tangent  t^  at  the  point  ^,  and  the  hexagon  approaches  the 
figure  composed  of  the  pentagon  P^I^P^P^I^  and  the  tangent  t^ 
to  the  conic  at  I^.  The  pairs  of  opposite  sides  are  ^j,  P^P ; 
I[P^,  I^Pq  ;  -^J^,  J^I^.  These  pairs  determine  collinear  points. 

Similarly,  ^  may  approach  7^  and  either  j^  approach  ^ 
or  J^  approach  ^,  yielding  an  inscribed  quadrilateral  and 
tangents  to  the  conic  at  two  of  the  vertices  of  the  quad- 
rilateral. A  third  case  is  that  in  which  ^  approaches  ^, 
^  approaches  ^,  and  ^  approaches  I^. 

In  each  case  the  propriety  of  extending  Pascal's  theorem,  and 
others,  to  limiting  cases  in  which  two  distinct  elements  are  allowed 
to  become  coincident  is  left  for  the  student's  consideration. 

In  the  case  of  a  circumscribed  hexagon,  if  one  side 
approaches  coincidence  with  a  second,  their  point  of  in- 
tersection approaches  a  limiting  position  at  the  point 
of  contact  of  the  second  side.  There  arise  out  of  the 
approach  of  sides  to  coincidence  a  number  of  limiting 
cases  of  Brianchon's  theorem  which  can  be  worked  out 
and  which  will  be  needed  from  time  to  time. 

Other  limiting  cases  of  these  propositions  are  those  in 
which  the  points  or  lines  of  the  figures  are  not  all  in  the 
finite  part  of  the  plane.  For  example,  one  or  two  vertices 
of  the  Pascal  hexagon  and  one  side  of  the  Brianchon 
hexagon  may  be  at  infinity.  Coincident  elements  and 
infinitely  distant  elements  may  be  present  in  one  hexagon. 


124  CONICS 

Problem,  conic  through  Five  Points 

126.    Griven  five  points  in  a  plane,  no  four  of  them  being 
collinear,  construct  the  conic  which  is  determined  by  them. 


Solution.    This  problem  admits  of  two  simple  solutions. 

1.  Method  based  on  a  projectivity. 

Let  the  given  points  be  I{,  J^,  ^,  ^,  ^,  Then  in  any 
chosen  direction  from  any  pomt,  as  J^,  there  can  be  found 
another  point  of  the  conic  which  is  not  collinear  with  two 
of  the  others.  Let  the  chosen  direction  be  along  the  line  jOj, 
and  let  the  point  to  be  found  be  called  P.  Draw  i^^, 
F,P,,  F,P,,  F^P^,  P^P,,  P^P,.  The  triads  of  lines  P,P„  P,P„  P,P, 
and  P^J^,  J^Jl,  I^P^  determine  the  projectivity  between  two 
flat  pencils  which  generate  the  required  conic. 

The  point  P  is  the  intersection  of  p^  and  its  correspond- 
ing line  of  the  pencil  whose  base  is  ^ ;  and  it  may  be  found 
by  the  method  used  in  §  39,  Case  1.  Draw  this  line  and 
produce  it  to  meet  j9j,  thus  determining  P. 

By  varying  the  position  of  the  line  p^  any  number  of 
additional  points  of  the  conic  may  be  found. 

Evidently  it  is  not  feasible  to  obtain  all  the  points  of  the  conic 
by  this  method,  nor  is  the  method  convenient  in  practice.  In  this 
respect  it  is  similar  to  the  method  of  plotting  in  analytic  geometry. 


CONIC  THROUGH  FIVE  POINTS  125 

2.  Method  based  on  PascaVs  theorem. 

Let  the  given  points  be  ^,  ^,  ^,  ^,  ^.  As  before,  an- 
other point  P  can  be  found  on  a  chosen  line  p^  that  passes 
tlirough  any  one  of  the  points,  as  J^. 


The  given  points  i^,  ^,  ^,  ^,  ^  and  the  point  P,  which 
is  to  be  found,  are  the  vertices  of  a  hexagon  inscribed  in 
the  conic  determined  by  the  five  given  points. 

Then  ^^,  ^^;  ^^,  P^P;  P^P^,  PJ^  intersect  on  the  Pascal 
line  of  this  hexagon.  Of  these  six  lines,  I^P  is  not  given 
and  PI^  is  the  given  line  py  The  Pascal  line  is  determined 
by  the  intersection  of  i^^,  ^^  and  that  of  ^^,  p^ 

Draw  the  Pascal  line  and  let  it  meet  P^I^  in  Q^.  Draw 
()2^.  This  line  Q^P,  must  coincide  with  the  line  J^P  and 
must  intersect  the  line  p^  in  the  required  point  P. 

Since  every  line  through  ij  determines  a  point  on  the 
conic,  it  is  possible  to  locate  any  number  of  points. 

This  method,  like  the  first  one,  bears  a  certain  resemblance  to 
the  method  of  plotting  in  analytic  geometry.  From  the  point  of 
view  of  convenience  it  is  decidedly  superior  to  the  first  method. 

The  student  will  observe  that,  since  the  conic  is  of  the 
second  order,  the  line  p^  cuts  it  in  one  and  only  one  point 
other  than  ^,  and  also  that  in  either  solution  of  the  prob- 
lem the  use  of  the  ruler  alone  is  sufficient. 


126  CONICS 

Problem.  Four  Pornxs  and  a  Tangent 

127.  G-iven  four  points  in  a  plane,  no  three  of  them  col- 
linear,  and  a  line  passing  through  one  and  only  one  of  these 
points,  construct  the  conic  which  passes  through  the  given 
points  and  at  one  of  them  is  tangent  to  the  given  line. 


Solution.  As  was  the  case  in  §  126,  there  are  two  simple 
methods  of  construction.  In  each  method  any  number  of 
additional  points  of  the  conic  may  be  found  by  determining 
where  the  conic  would  be  cut  by  lines  which  pass  through 
one  of  the  points. 

1.  Method  based  on  a  projectivity. 

Let  the  given  points  be  J^,  I^,  P^,  P^,  and  let  the  given 
line  be  t^  passing  through  the  point  -^.  Draw  any  line  p^ 
through  the  point  !{.  Join  i^  to  each  of  the  points  P^,  P^,  -^, 
and  join  P^  to  each  of  the  points  P^,  I^. 

The  triads  ^^,  ^^,  P^P^  and  t^,  P^P^,  P^P^  determine  a 
projectivity  between  the  flat  pencils  whose  bases  are  I{,  J^, 
and  the  required  conic  is  the  locus  of  the  intersections  of 
corresponding  lines  of  the  projective  pencils. 

The  line  through  J^  which  corresponds  to  p^  of  the  pencil 
whose  base  is  J^  can  be  determined  by  the  method  used  in 
§  39,  Case  1,  and  P,  the  intersection  of  this  line  with  jOj,  is 
the  point  required. 

By  varying  the  position  of  p^  any  number  of  points  of 
the  conic  may  be  found. 


FOUR  POINTS  AND  A  TANGENT 

2.  Method  based  on  PascaVs  theorem. 


127 


Let  P  be  the  point  on  p^  that  is  to  be  found.    It  is 
determined  if  the  direction  of  P.P  can  be  determined. 


The  pentagon  I^P^Pr^P  is  inscribed  in  the  required 
conic,  and  the  line  t^  is  tangent  to  the  conic  at  J^. 

The  intersections  of  ^^,  ^^;  t^,  P^P ;  and  ^i^,  PP^ 
(or  jt?j)  are  on  the  Pascal  line. 

Produce  P^I^  and  P^Pr^  to  meet  at  Q^,  and  produce  P^P^ 
and  p^  to  meet  in  ^3.  Draw  the  Pascal  line.  Let  t^  meet 
this  line  in  Q^ ;  join  ^  and  Q^. 

Then  the  lines  I^Q^  ^^nd  i^P  are  coincident  and  the 
intersection  of  P,Q^  and  jPj  is  the  required  point  P. 


Problem.  Three  points  and  Two  Tangents 

128.  Given  in  a  plane  three  noncolUnear  points  and  two 
lines,  each  of  which  passes  through  one  and  only  one  of  the 
given  points,  construct  the  conic  which  passes  through  the  three 
given  points  and  at  each  of  two  of  them  is  tangent  to  the 
given  line  through  that  point. 

The  solution  is  left  for  the  student.  It  should  be  effected  by  two 
methods,  as  in  the  two  preceding  theorems.  As  in  the  other  prob- 
lems the  second  method  is  to  be  preferred  for  practical  reasons. 

An  appreciation  of  the  superior  convenience  of  the  second  method 
is  best  secured  by  making  the  actual  construction  necessary  for  find- 
ing by  the  first  method  the  line  through  P^  which  corresponds  to  p^ 
of  the  first  pencil. 


128  CONICS 

Problem.  Three  Tangents  and  Two  Points 

129.  Given  three  nonconcurrent  lines  in  a  plane,  and  on 
each  of  two  of  these  lines  a  point  which  is  not  on  any  other 
of  the  three,  construct  the  conic  which  is  tangent  to  each  of 
the  given  lines  and  has  each  of  the  given  points  as  the  point 
of  contact  of  the  given  line  on  which  it  lies. 

Of  what  problem  is  this  the  dual?  The  solution  is  left  for 
the  student. 

Problem.  Four  Tangents  and  One  Point 

130.  Given  four  lines  in  a  plane,  no  three  of  them  con- 
current,  and  a  point  on  otie  but  not  on  two  of  them,  construct 
the  conic  which  is  tangent  to  each  of  these  lines  and  has  the 
given  point  as  the  poitit  of  contact  of  the  given  line  on  which 
it  lies. 

Of  what  problem  is  this  the  dual?  The  solution  is  left  for 
the  student. 

Problem.  Five  Tangents 

131.  Given  five  lines  in  a  plane,  no  four  of  them  con- 
current, construct  the  conic  which  is  tangent  to  each. 

Of  what  problem  is  this  the  dual?  The  solution  is  left  for 
the  student. 

Problem,  constructing  a  Tangent 

132.  Given  five  or  more  points  of  a  conic,  construct  the 
tangent  to  the  conic  at  any  one  of  these  points. 

The  solution  is  left  for  the  student. 

Problem.  Finding  a  Point  of  contact 

133.  Given  five  or  more  tangents  to  a  conic,  determine  by 
construction  the  poifit  of  contact  of  any  one  of  these  tangents. 

The  solution  is  left  for  the  student. 


PROBLEMS  OF  CONSTRUCTION      129 

Exercise  27.    Problems  of  Construction 

1.  If  two  projective  flat  pencils  generate  a  circle,  they  are 
congruent. 

2.  Using  the  result  in  Ex.  1,  find  any  number  of  additional 
points  of  a  circle  when  three  of  its  points  are  given. 

3.  Find  any  number  of  points  of  a  circle  when  two  of  its 
points  and  the  tangent  at  one  of  them  are  given. 

4.  Solve  the  problem  in  §  126  when  one  of  the  five  points 
is  at  infinity  in  a  given  direction. 

5.  Solve  the  problem  in  §  126  when  two  of  the  points  are 
at  infinity  in  given  directions. 

6.  Solve  the  problem  in  §  127  when  the  given  line  is  at 
infinity. 

7.  Solve  the  problem  in  §  127  when  one  of  the  four,  points 
is  at  infinity  in  a  given  direction. 

8.  Solve  the  problem  in  §  128  when  one  of  the  points  is 
at  infinity  in  a  given  direction  and  the  tangent  at  that  point 
is  given  to  be  the  line  at  infinity. 

9.  Solve  the  problem  in  §  129  when  the  two  given  points 
are  at  infinity. 

10.  Solve  the  problem  in  §  131  when  one  of  the  five  given 
lines  is  the  line  at  infinity. 

11.  Solve  the  problem  in  §  132  when  the  point  at  which  the 
tangent  is  to  be  constructed  is  at  infinity  in  a  given  direction. 

12.  Solve  the  problem  in  §  133  when  the  given  tangent 
whose  point  of  contact  is  to  be  found  is  the  line  at  infinity. 

y~  13.  If  a  parallelogram  is  inscribed  in  a  conic,  the  tangents 
to  the  conic  at  the  vertices  form  a  parallelogram  circumscribed 
about  the  conic. 

•^14.  If  P^,  P^,  Pg,  P^,  P5  are  fixed  points  and  P  moves  on  the 
conic  determined  by  them,  find  the  envelope  of  the  Pascal  line 
of  the  hexagon  P^I^P^P^PrP- 


130  CONICS 


Theorem,  involution  on  complete  Quadrangle 

134.  If  a  straight  line  cuts  all  the  sides  of  a  complete 
quadrangle  but  does  not  pass  through  any  vertex,  it  cuts  the 
three  pairs  of  opposite  sides  of  the  quadrangle  in  conjugate 
points  of  an  involution. 


Proof.  Let  a  straight  line  p  cut  the  pairs  of  opposite 
sides  of  the  complete  quadrangle  whose  vertices  are  I{,  I^, 
^,  i^  in  A,  A';  B,  B';  C,  C;  and  let  il/,  iV,  0  be  the  diagonal 
points  of  the  quadrangle. 

Then  range  ^^^'C^  flat  pencil  J^(MF^A'F^^ 

-^  range  MI^A'Pj^ 
-^  flat  pencil  JJ  (MP^A'F^) 
■^  range  AC'A'B'. 

But  range  ^5^'C^  range  A'CAB.  §  23 

Hence        range  AC'A'B'  -^  range  A'CAB. 

Accordingly,  A,  A';  B,  B';  C,  C  are  conjugate  points  of 
an  involution  on  p.  §  65 

This  theorem  is  auxiliary  to,  and  is  in  fact  a  special  case  of,  an 
interesting  and  important  theorem  which  was  first  established  by 
the  French  geometer  Girard  Desargues  (1593-1662). 

Desargues's  theorem  offers  another  line  of  apjiroach  to  some  of 
the  preceding  constructions  and  to  other  similar  problems.  In 
particular,  on  page  133,  it  is  applied  to  the  solution  of  §  126. 


DESARGUES'S  THEOREM 
Theorem.  Desargues's  Theorem 


131 


135.  If  a  complete  quadrangle  is  inscribed  in  a  conic,  and 
if  a  straight  line  cuts  the  conic  in  ttvo  points  distinct  from 
each  other  and  from  the  vertices  of  the  quadrangle,  these  two 
points  form  a  conjugate  pair  of  the  involution  of  points  on 
the  line,  which  is  determined  hy  the  intersections  of  the  line 
with  the  pairs  of  opposite  sides  of  the  quadrangle. 


Proof.  Let  the  complete  quadrangle  P^I^I^Il  be  inscribed 
in  a  conic,  and  let  a  line  p  which  does  not  pass  through 
any  of  the  four  vertices  cut  the  conic  in  P,  P'  and  the 
pairs  of  opposite  sides  m.  A,  A';  B,  B' ;  C,  C'. 

Then  range  P^P'^- flat  pencil  i^(P^P'^) 

-flat  pencil  ^  (P^P'i^)  -  range  PA'P'B'. 
But  range  P^'P'^' -^  range  P'B'PA'.  §  23 

Therefore         range  PPP'^  -  range  P'P'PJ'. 

Hence  P,  P';  A,  A';  B,  B'  are  conjugate  points  of  an 
involution  on  p  determined  by  the  pairs  A,  A' ;  B,  B'. 

The  involution  formed  by  the  intersections  of  p  with  the 
pairs  of  opposite  sides  of  the  quadrangle  I^I^P^P^  is  also 
determined  by  the  pairs  of  points  A,  A';  B,  B'. 

Accordingly,  P,  P'  are  conjugate  points  of  the  involution 
of  points  determined  by  the  intersections  of  the  line  p  with 
the  pairs  of  opposite  sides  of  the  quadrangle  P^P^P^P^. 


132  CONICS 

136.  Restatement  of  Desargues's  Theorem.  It  should  be 
noted  that  many  conies  pass  through  the  four  points  ij,  i^, 
i^,  i^  and  that  to  each  of  such  conies  Desargues's  theorem 
apphes.  Moreover,  it  should  be  remembered  that  the  pairs 
of  lines  7^-^,  ^/^;  I^F^,  I^P^  are  degenerate  conies  through 
the  four  pomts.  Hence  Desargues's  theorem  is  capable  of 
restatement  as  follows : 

The  infinitely  many  conies,  including  pairs  of  lines,  wJiich 
pass  through  four  given  coplanar  points,  no  three  of  which  are 
collinear,  determine  on  any  line  which  intersects  them  (hut  does 
not  pass  through  any  one  of  the  points^  infinitely  many  pairs 
of  points  of  an  involution. 

137.  Corollary.  If  the  involution  determined  by  the 
conies  is  hyperbolic,  two  of  the  conies  which  pass  through  the 
four  points  touch  the  straight  line;  if  it  is  elliptic,  no  conic 
through  the  four  points  is  tangent  to  the  straight  line. 

Exercise  28.    Application  of  Desargues's  Theorem 

1.  What  sort  of  involution  is  determined  upon  a  side  of  the 
diagonal  triangle  of  the  quadrangle  mentioned  in  the  theorem 
of  §  135  ? 

2.  Test  the  validity  of  the  proof  of  Desargues's  theorem  when 
it  "is  applied  to  a  line  through  one  of  the  given  points,  say  P^. 

Given  four  points  in  a  plane,  no  three  of  which  are  col- 
linear,  show  how  to  draw  a  straight  line  subject  to  each  of  the 
following  conditions : 

3.  There  shall  be  two  conies  passing  through  the  four  points 
and  tangent  to  the  line. 

4.  There  shall  be  one  conic  passing  through  the  four  points 
and  tangent  to  the  line. 

5.  There  shall  be  no  conic  such  as  described  in  Ex.  4. 


DESAKGUES'S  THEOREM  133 

Problem.  Conic  through  Five  Points 

138.  G-iven  five  points,  no  four  of  which  are  coUinear,  con- 
struct the  conic  ivhich  is  determined  hy  them. 


Solution.    Let  us  consider  two  cases. 

1.  No  three  of  the  five  given  jyoinis  are  coUinear. 

Let  the  five  points  be  i^,  ^,  i*,  i^,  7^.  Draw  JJi^,  ^i^, 
i^i^,  P^J^,  F^F^,  J^I^,  and  any  transversal  p^  through  F^. 

It  is  now  required  to  find  the  point  F  in  which  this  hue 
again  cuts  the  conic  determined  by  the  five  given  points. 

The  points  Ap  A^,  B^  B^,  in  which  p.^  is  cut  by  the  hues 
J^/^,  i^i^,  F^F^,  B^I^,  determine  an  invohition  in  which  ij 
and  the  required  point  F  are  corresponding  points. 

Hence  the  point  F  can  be  determined  from  the  pro- 
jectivity  between  the  ranges  in  the  involution.  For  in- 
stance, the  three  known  points  A^  B^  A^  and  the  required 
point  F  are  projective  with  the  four  known  points  A^,  B^, 
Jp  Fy  Various  special  devices  for  finding  F  based  upon 
the  method  of  §  39,  Cases  1  and  2,  can  be  found. 

By  varying  the  position  of  the  line  p^  any  number  of 
points  on  the  conic  can  be  found. 

2.  Three  of  the  five  given  points  are  coUinear. 

In  this  case  the  required  conic  is  a  pair  of  straight  lines, 
one  the  hue  through  the  three  points  and  the  other  the  line 
through  the  other  two  pomts. 


134  CONICS 

Problem.  Position  of  Self-corresponding  Elements 

139.  Given  two  superposed  projective  one-dimensional  prime 
fortnSy  construct  the  position  of  the  self-corresponding  elements. 


Solution.  If  the  superposed  prime  forms  are  not  ranges, 
it  is  possible  by  operations  of  projection  and  of  section  to 
obtain  from  them  two  superposed  projective  ranges.  Hence 
it  is  necessary  to  solve  the  problem  only  for  the  case  in 
which  the  prime  forms  are  ranges. 

Let  A^B^C^  and  A^B^C^  be  two  triads  of  corresponding 
points  of  superposed  projective  ranges  on  a  base  p. 

Describe  any  circle  coplanar  with  the  line  p.  Join  any 
point  F  of  the  circle  to  each  of  the  six  given  points,  and  let 
these  lines  cut  the  circle  again  in  A[,  B[,  Cj  and  A^-,  B^-,  Cg. 
Join  A[  to  A'^,  B'^,  C^,  and  A'^  to  B[,  C{. 

Then  flat  pencil  A'^  (A[B[C[  -  •  •) 

-  flat  pencil  P(A[B[C[  •  .  .)  §  53 
-range  A^B^C^  . -- 

-^  range  ^2^2  ^2  *  *  * 

-  flat  pencil  P  (A'^B'^C;^ . . .) 

-  flat  pencil  A[  {A'^B'^C^  •  •  •)• 

But  the  flat  pencils  A'^  iA[B[C[  -  •  •),  A  iAK^^'i '  *  0  ^ave 
a  self -corresponding  element,  and  hence  are  perspective. 


CONSTRUCTIONS  OF  THE  SECOND  ORDER    135 

Let  X'  be  a  point,  if  there  be  any,  in  which  p\  the  axis  of 
perspectivity,  cuts  the  circle,  and  let  PX'  meet  p  at  X. 

In  the  four  flat  pencils  previousl)'-  mentioned  the  corre- 
sponding lines  are  A\X',  PX',  A'^X',  PX',  and  hence  in  the 
superposed  pencils  whose  bases  are  at  P,  PX'  is  a  self- 
corresponding  line.  Therefore  X  is  a  self-corresponding 
point  of  the  ranges  on  p. 

Conversely,  it  is  true  that,  corresponding  to  each  self- 
corresponding  point  of  the  ranges  on  the  line  p,  there  is  an 
intersection  of  the  line  p'  with  the  circle. 

Hence,  to  find  the  self -corresponding  points  on  p  we  join 
P  to  the  intersections  of  p'  and  the  circle,  and  produce 
these  lines  to  intersect  p.  There  may  be  no,  one,  or  two 
intersections  with  p,  and  each  of  these  intersections  is  a 
self-corresponding  point. 

140.  Constructions  of  the  Second  Order.  All  constructions 
made  before  §  139  were  effected  wholly  by  the  use  of 
straight  lines,  and  at  every  stage  the  results  were  uniquely 
determinate ;  that  is,  all  the  problems  had  one  and  only 
one  solution.  If  the  solutions  of  the  problems  analogous 
to  these  constructions  are  effected  by  the  methods  of 
analytic  geometry,  it  is  found  that  only  equations  of  the 
first  degree  are  used.  For  this  reason  these  and  similar 
problems  are  said  to  be  of  the  first  order. 

Beginning  with  §  141,  attention  will  be  given  to  prob- 
lems whose  solutions  by  the  method  of  analytic  geometry 
would  involve  the  use  of  at  least  one  equation  of  the  second 
degree,  as  in  §  139.  Correspondingly,  each  construction  will 
require  the  use  (at  least  once)  of  a  curve  of  tlie  second 
order,  and  for  simplicity  the  circle  will  be  taken. 

The  problem  in  §  139  furnishes  a  basis  for  others,  and  hence  has 
been  deferred  from  its  most  natural  jilace,  which  was  in  connection 
with  the  treatment  of  superposed  projective  forms  in  Chapter  VII. 


136  CONICS 

Problem,  intersections  of  a  Line  and  a  Conic 

141.  Given  any  of  the  sets  of  elements  mentioned  in  §117 
as  determining  a  conic,  construct  the  intersections  (if  there  are 
any)  of  the  conic  with  a  given  straight  line  p  in  its  plane. 

Solution.  If  the  set  of  elements  is  not  five  points,  by 
means  of  §§  127-133  find  five  points  P^,  Pg,  i^,  i^,  P^  on  the 
conic.  Join  any  two  of  the  points,  as  ij  and  j^,  to  F^,  P^,  i^, 
and  let  these  lines  meet  the  line  p  in  P^,  P^,  P^  and  i^",  i^",  Jp". 

These  triads  determine  superposed  projective  ranges  on  jj. 
By  the  method  of  §  139  find  the  self -corresponding  points 
(if  there  are  any)  of  these  ranges. 

Since  these  self-corresponding  pomts  are  common  to 
corresponding  lines  of  the  projective  flat  pencils  whose 
bases  are  ij  and  1^,  they  are  on  the  conic.  Moreover,  they 
are  the  only  points  of  p  which  are  on  the  conic.  There 
may,  therefore,  be  two,  one,  or  no  intersections. 

Problem.  Tangents  from  a  Point 

142.  Given  any  of  the  sets  of  elements  mentioned  in  §  117 
as  determining  a  conic,  construct  the  tangents  (if  there  are  any) 
to  the  conic  from  a  given  point  P  in  its  plane. 

Solution.  If  the  set  of  given  elements  is  not  five  tangents 
to  the  conic,  by  means  of  §§  127-133  find  five  tangents 
^1'  ^2'  ^3'  ^4'  ^5  ^®  ^'^®  conic.  The  tangents  ^3,  t^,  t^  cut  t^,  t^ 
in  triads  of  points  which,  being  joined  to  P,  determine  a 
projectivity  between  flat  pencils  whose  base  is  P. 

Find  the  self-corresponding  lines  of  these  pencils.  Any 
such  line  passes  through  P  and  also  joins  corresponding 
points  of  the  projective  ranges  on  t^,  t^  which  serve  to 
generate  the  conic.  Hence  the  line  is  tangent  to  the  conic. 
The  number  of  these  lines  is  two,  one,  or  none. 


PROBLEMS  OF  CONSTRUCTION 
Problem.  Four  Points  and  a  Tangent 


137 


143.  Construct  a  conic  which  shall  pass  through  four  given 
points,  no  three  of  which  are  collinear,  and  shall  be  tangent  to 
a  given Jine  that  does  not  contain  any  of  the  points. 


Solution.  Let  the  given  points  be  ij,  i^,  i^,  i^,  and  let 
the  given  line  be  ty  Let  t^  cut  the  lines  J^i^,  F^P^  in  A^  A^ 
and  cut  the  lines  P^P^,  i^-?^  in  By,  B^. 

Find  the  self-corresponding  points  of  the  involution  on  t^ 
which  is  determined  by  these  pairs  of  pomts.  Through  the 
four  points  and  any  self-corresponding  point  i^  construct 
a  conic  (§  138).  This  conic  is  tangent  to  fj  at  i^.  For  if 
it  cuts  t^  in  a  second  point  i^,  then  the  point  i^  is  not 
self-corresponding,  and  this  is  contrary  to  fact. 

Hence,  for  every  self -corresponding  point  of  the  involu- 
tion on  t^  one  conic  can  be  constructed. 

There  may  be  no,  one,  or  two  self-corresponding  points 
(§  139).    Hence  no,  one,  or  two  conies  may  be  constructed. 

Theorem.  Four  Points  and  a  Tangent 

144.  The  number  of  conies  which  pass  through  four  given 
points,  no  three  of  ivhich  are  collinear,  and  are  tangent  to 
a  given  straight  line  which  does  not  pass  through  any  of  the 
points,  is  none,  'one,  or  two. 

The  proof  is  left  for  the  student. 


138  CONICS 

Problem.  Four  Tangents  and  a  Point 

145.  Construct  a  conic  ivhich  shall  be  tangent  to  each 
of  four  given  straight  lines,  no  three  of  which  are  concur- 
rent, and  which  shall  pass  through  a  given  point  exterior  to 
the  lines. 

This  problem  is  the  dual  of  §  143  and  may  be  solved  as  such. 
The  solution  is  left  for  the  student.  Likewise,  a  theorem  dual  to 
§  144  results  from  the  proof  of  the  construction. 

146.  Special  Case  of  Desargues's  Theorem.  To  complete 
a  set  of  constructions  which  include  §§  126, 129-133,  143, 
and  145,  two  others  are  necessary,  and  these  are  given  in 
§§  148  and  149.  In  order  to  solve  these  two  problems  special 
cases  of  Desargues's  theorem  (and  its  dual)  may  be  used. 

Instead  of  the  four  distinct  pomts  of  the  conic  con- 
sidered in  Desargues's  theorem,  let  the  first  and  second 
points  move  up  to  coincidence,  and  also  let  the  third  and 
fourth  points  move  up  to  coincidence.  Then  the  line  join- 
ing the  first  and  second  points  and  that  joining  the  third 
and  fourth  points  become  tangents  to  the  conic.  Also  the 
lines  joining  the  first  and  third  points,  the  second  and 
fourth  points,  the  first  and  fourth  points,  and  the  second 
and  third  points  move  into  coincidence  upon  the  chord  of 
contact  of  the  two  tangents  mentioned. 

Theorem.  Special  Form  of  Desargues's  Theorem 

147.  Two  straight  lines  and  the  conies  which  are  tangent 
to  them  at  two  given  points  intersect  a  given  line  that  does 
not  pass  through  either  of  these  points  in  pairs  of  points  of  an 
involution,  one  of  the  self-corresponding  points  of  which  is  the 
intersection  of  the  given  line  with  the  chord  of  contact. 

The  proof  is  left  for  the  student. 


PROBLEMS  OF  CONSTRUCTION  139 

Problem.  Three  Points  and  Two  Tangents 

148.  Construct  a  conic  which  shall  pass  through  each  of 
three  given  noncollinear  points  and  he  tangent  to  each  of  two 
given  lines  that  do  not  pass  through  any  of  the  points. 


Solution.  Let  the  points  be  i^,  ^,  i^,  and  let  the  lines 
be  ^1,  ^2-  Let  t^,  t^  cut  the  line  P^P^  in  A^,  A^  and  the  line 
P^P^  in  i?j,  i>2.  We  shall  first  find  the  points  of  contact  of 
<i,  <2  with  the  conic. 

Find  the  self-corresponding  points  (if  there  are  any)  of 
the  involutions  determined  by  i^,  P^  and  Jj,  A^  and  by 
^,  P^  and  B^,  B^  respectively.  Let  the  line  through  JT/^, 
one  of  the  first  of  these,  and  M^,  any  one  of  the  second,  cut 
^2  in  P^  and  t^  in  P^.  Pass  a  conic  through  i^,  and  tangent 
to  t^  and  ^2  at  P^  and  P^  respectively  (§  128). 

Since  the  point  corresponding  to  P^  in  the  involution 
on  A^A^  is  completely  determined  by  Mj^  and  the  pair  of 
points  Jj,  ^2'  it  follows  that  this  conic  must  pass  through 
i^  (§  147),  for  the  involution  determined  on  P^I^  by  conies 
tangent  at  ^  and  P^  to  ^g  ^^^  h  ^^  ^^^^  determined  by 
a  self-corresponding  point  and  the  pair  of  points  A^,  A^. 
Similarly,  this  conic  can  be  shown  to  pass  through  i^. 

Hence,  for  every  possible  pair  of  points  ^,  i^  one  conic 
may  be  constructed. 

There  may  be  no,  one,  two,  or  four  pairs  of  points,  as 
i^,  i^  (§  144),  and  for  each  a  conic  may  be  constructed. 

PG 


140  CONICS 

Theorem.  Three  Points  and  Two  Tangents 

149.  The  number  of  conies  which  pass  through  three  given 
noncollinear  points  and  which  are  tangent  to  two  given  lines 
that  do  not  contain  any  of  the  points  is  none,  one,  two,  or  four, 
as  the  case  may  he. 

The  student  should  write  out  the  proof,  which  is  essehtially  that 
of  §  148. 

PROBLEM.  Three  Tangents  and  Two  Points 

150.  Construct  a  conic  which  shall  he  tangent  to  three  given 
nonconcurrent  lines  and  shall  pass  through  two  given  points 
which  are  exterior  to  the  lines. 

The  student  should  write  out  the  solution,  wliich  is  simply  the 
dual  of  that  of  §  148. 

Exercise  29.    Review 

*  1.  If  the  sides  of  an  angle  of  constant  size  rotating  about 
a  fixed  vertex  intersect  respectively  two  fixed  lines,  the  line 
joining  these  intersections  envelops  a  conic.  '■(r-_    i>i^*lz/*i 

2.  Two  vertices  of  a  variable  triangle  move  along  two  fixed 
lines,  and  the  three  sides  respectively  pass  through  three  fixed 
collinear  points.    Eind  the  locus  of  the  third  vertex. 

3.  Consider  Ex.  2  for  the  case  in  which  the  three  fixed 
points  are  not  collinear. 

4.  If  two  triangles  are  in  plane  homology,  the  intersections 
of  the  sides  of  one  triangle  with  the  noncorresponding  sides 
of  the  other  lie  on  a  conic. 

5.  State  Pascal's  theorem  for  the  case  in  which  the  first 
and  second,  the  third  and  fourth,  and  the  fifth  and  sixth 
vertices  have  become  coincident. 

6.  The  complete  quadrilateral  formed  by  four  tangents  to 
a  conic,  and  the  complete  quadrangle  formed  by  their  four 
points  of  contact,  have  the  same  diagonal  triangle. 


EEVIEW  EXERCISES  141 

7.  If  a  variable  quadrangle  P^P^P^P^  inscribed  in  a  conic 
has  as  fixed  points  P^,  P^,  and  the  intersection  of  P^P^,  P^P^,  the 
other  vertices  of  its  diagonal  triangle  move  along  the  same 
fixed  straight  line. 

8.  If  Pg,  Pg  are  fixed  points  on  a  given  conic,  and  if  P  is 
a  moving  point,  as  P  moves  along  the  conic  the  Pascal  line 
of  the  hexagon,  consisting  of  the  triangle  PPgPg  and  the 
tangents  to  the  conic  at  the  points  P^,  P^,  Pg,  envelops  a  conic. 

9.  If  Pj,  Pg,  P^  are  fixed  vertices  of  a  complete  quadri- 
lateral whose  fourth  vertex  P  moves  along  a  given  conic 
through  Pj,  Pg,  P_j,  all  the  vertices  of  the  diagonal  triangle  trace 
straight  lines  and  all  the  sides  pass  through  fixed  points. 

10.  If  the  points  P^  and  P^  trace  superposed  projective 
ranges  on  the  base  AB  of  a  fixed  triangle  ABC,  if  Pj  is  a  fixed 
point  not  on  any  side  of  the  triangle,  if  P^P^  meets  A  C  in  P^, 
and  if  P^Pg  meets  BC  in  Pg,  find  the  locus  of  P,  the  inter- 
section of  AP,  and  JSP.. 

5  4 

11.  In  Ex.  10  find  the  envelope  of  P^P^ 

12.  State  Desargues's  theorem  for  the  case  in  which  a  pair 
of  the  four  given  coplanar  points  become  coincident. 

13.  State  Desargues's  theorem  for  the  case  in  which  two 
pairs  of  the  given  coplanar  points  become  coincident. 

14.  Three  sides,  AB,  AD,  CD  respectively,  of  a  variable 
quadrangle  inscribed  in  a  given  conic  pass  through  three  given 
points  of  a  line.    Find  the  envelope  of  BC. 

15.  Extend  Ex.  14  to  the  case  of  a  simple  inscribed  polygon 
having  2  n  sides. 

16.  From  the  data  of  §  127  construct,  by  means  of  §  147, 
tangents  at  additional  points  of  the  conic. 

17.  Prove  the  dual  of  §  149,  namely,  that  the  number  of 
conies  which  can  be  constructed  under  the  conditions  of  §  150 
is  none,  one,  two,  or  four. 

18.  Solve  the  dual  of  §  139  for  the  plane. 


142  CONICS 

19.  If  the  lines  2^1  and  p^  are  drawn  through  the  vertices  P^ 
and  Pj  respectively  of  a  given  quadrangle,  the  conies  which 
pass  through  the  vertices  of  the  quadrangle  determine  perspec- 
tive ranges  on  p^  and  j-j^. 

20.  If  the  lines  p^  and  p^  are  drawn  through  the  vertex  P^ 
of  a  given  quadrangle,  the  system  of  conies  which  pass  through 
the  vertices  of  the  quadrangle  determine  projective  ranges 
on  p^  and  p^. 

21.  State  and  prove  the  dual  of  Ex.  19  for  the  plane. 

22.  Construct  a  conic  which  shall  pass  through  two  given 
points  Pj  and  P^,  shall  be  tangent  to  a  given  line  t^  at  the 
point  Pg,  and  shall  be  tangent  to  a  second  given  line  t^. 

Apply  Ex.  12  for  the  line  t^.  Find  the  self-corresponding  points  of 
the  involution. 

23.  Construct  a  conic  which  shall  be  tangent  to  a  given 
line  t^  at  the  point  P^,  to  t^  at  P^,  and  to  t^. 

24.  Consider  the  problem  of  §141  for  the  case  in  which  the 
given  line  is  the  line  at  infinity. 

25.  Consider  the  problem  of  §143  for  the  case  in  which  the 
given  line  is  the  line  at  infinity. 

26.  Consider  the  problem  of  §  148  for  the  case  in  which  one 
of  the  given  lines  is  the  line  at  infinity. 

27.  Solve  the  dual  of  Ex.  22  for  the  plane. 

28.  Construct  a  triangle  which  shall  be  inscribed  in  a  given 
triangle  and  have  its  sides  pass  through  three  given  points. 

Observe  that  if  a  triangle  has  two  vertices,  as  required,  but  not  the 
third,  the  sides  through  the  latter  cut  a  side  of  the  given  triangle  in 
corresponding  points  of  superposed  projective  ranges. 

29.  Construct  a  triangle  which  shall  be  inscribed  in  a  given 
conic  and  have  its  sides  pass  through  three  given  points. 

30.  If  a  conic  can  be  described  through  the  six  vertices  of 
two  given  triangles,  another  conic  can  be  described  which  shall 
be  tangent  to  the  six  sides  of  the  two  given  triangles. 


CHAPTER  XI 

CONICS  AND  THE  ELEMENTS  AT  INFINITY 

151.  Classification  of  Conies.  In  the  discussion  of  conies 
in  the  preceding  chapter  no  classification  was  made,  nor 
was  any  account  taken  of  the  fact  that  on  certain  occasions 
the  term  straight  line  may  mean  "  straight  line  at  infinity  " 
and  the  term  point  may  mean  "  point  at  infinity."  These 
considerations  can  be  associated  very  advantageously. 


Ellipse  Pakabola  Hyperbola 

In  projective  geometry,  conies  are  classified  by  means  of 
their  relations  to  t^Q  line  at  infinity.  This  line,  like  any 
other,  may  intersect  a  conic  in  no,  one,  or  two  points,  and 
hence  conies  are  divided  into  three  classes  as  follows : 

1.  Ellipses,  or  conies  that  do  not  intersect  the  line  at 
infinity. 

2.  Parabolas,  or  conies  that  intersect  the  line  at  infinity 
in  one  point  (or  are  tangent  to  the  line  at  infinity). 

3.  Hyperbolas,  or  conies  that  intersect  the  line  at  infinity 
in  two  distinct  points. 

While  these  conies  are  familiar  to  the  student  from  his  work  in 
analytic  geometry,  the  study  of  conies  will  now  be  considered  from 
a  different  point  of  view. 

143 


144    CONICS  AND  THE  ELEMENTS  AT  INFINITY 

152.  Elements  at  Infinity.  In  the  interpretation  of  the 
results  already  obtamed,  in  so  far  as  the  ellipse  is  con- 
cerned, it  will  be  seen  that  the  expressions  point  on  the  curve 
and  tangent  to  the  curve  always  mean  a  point  and  a  line  in 
the  finite  part  of  the  plane. 

On  the  other  hand,  in  connection  with  the  parabola,  one 
and  only  one  tangent,  and  one  and  only  one  point  of  the 
curve  (the  point  of  contact  of  that  tangent),  may  be  taken 
to  be  at  infinity. 

In  the  case  of  the  hyperbola  there  are  two  points  on  the 
curve  which  are  at  infinity,  but  the  line  at  infinity  is  not 
a  tangent.  At  each  of  the  infinitely  distant  points  of  the 
curve  there  is,  however,  a  tangent  which  has  no  infinitely 
distant  point  except  its  point  of  contact. 

It  follows  that  in  the  cases  of  the  parabola  and  hyperbola 
the  interpretations  of  the  theorems  of  Pascal  and  Brianchon 
and  of  similar  theorems  obtained  by  the  methods  of  pro- 
jective geometry  vary  according  as  all  or  only  part  of  the 
elements  are  assumed  to  be  in  the  finite  part  of  the  plane. 

In  the  light  of  the  procedure  indicated,  the  results  which  have 
been  obtained  are  capable  of  restatements  which  vary  for  the  three 
types  of  conies,  but  which  have  a  great  interest,  because  they  bring 
these  results  into  clearer  relation  to  those  obtained  by  the  methods 
of  analytic  geometry. 

153.  Asymptote.  A  line,  not  the  line  at  infinity,  which 
is  tangent  to  a  conic  at  an  infinitely  distant  point  is  called 
an  asymptote. 

In  this  figure  a  is  an  asymptote. 

Every  hyperbola  has,  then,  two  asymp- 
totes, and  the  other  conies  have  none, 
though  sometimes  the  parabola  is  said 
to  have  the  line  at  infinity  as  an  asymptote.  This  latter  form  of 
statement  is  convenient  when  geometry  is  treated  algebraically, 
but  it  will  not  be  adopted  in  this  text. 


ELEMENTS  AT  INFmiTY  145 

154.  Special  Interpretations.   As  indicated  in  §  152,  each 

of  the  results  that  have  been  derived  for  the  conies  should 
be  examined  for  interpretations  based  upon  the  rela- 
tions of  the  elements  at  infinity  to  the  three  types  of 
conies.  The  great  variety  of  results  that  can  be  obtained 
prevents  a  systematic  and  detailed  reexamination  in  this 
place  of  all  the  theorems  and  constructions  that  have 
been  derived.  A  few  of  these  will  be  obtained,  but  for  the 
most  part  their  derivation  must  be  left  to  the  student,  a 
work  which  will  prove  both  interesting  and  profitable. 

Of  the  elements  (points  and  lines)  which  determine  a 
conic  not  more  than  two  points  and  not  more  than  one 
line  may  be  at  infinity,  except  in  the  limiting  case  of 
coincident  points  or  coincident  tangents.  The  existence 
of  one  infinitely  distant  point  on  a  conic  determines  that 
the  curve  is  not  an  ellipse,  and  the  existence  of  two  such 
points  determines  that  the  curve  is  a  hyperbola.  Similarly, 
the  tangency  of  the  line  at  infinity  to  the  curve  determines 
it  to  be  a  parabola. 

On  the  other  hand,  when  all  the  given  determining 
elements  are  in  the  finite  part  of  the  plane,  the  conic  may 
prove  to  be  of  any  one  of  the  three  types.  The  determi- 
nation of  the  character  of  the  conic  of  which  certain  ele- 
ments are  given  is  a  particularly  interesting  case.  It  is  the 
problem  of  §  141  as  modified  in  Ex.  24,  page  142. 

In  view  of  what  is  said  above,  we  shall  now  restate 
the  important  theorem  of  §  116. 

In  each  case  the  student  should  draw  the  figure  and  satisfy  him- 
self that  the  statement  is  correct  and  that  it  is  a  special  case  of  one 
of  the  corresponding  statements  in  §§  116  and  117.  He  should  also 
supplement  the  results  here  set  forth  by  the  others  which  can  be 
obtained  if  a  thorough  examination  of  the  theorem  is  made  for  its 
various  interpretations. 


146    CONICS  AND  THE  ELEMENTS  AT  INFINITY 

Theorem,  a  conic  Determined 

155.  1.  There  is  one  and  only  one  conic  (^parabola  or 
hyperbola)  zvhich  passes  through  four  points  in  the  finite 
part  of  a  plane  and  one  infinitely  distant  point  in  a  speci- 
fied direction. 

This  follows  from  the  theorem  stated  in  §  116,  No.  1. 

2.  Tliere  is  one  and  only  one  hyperbola  which  passes  through 
three  points  in  the  finite  part  of  a  plane  and  has  given  direc- 
tions for  its  asymptotes. 

This  follows  from  the  theorem  stated  in  §116,  No.  1. 

3.  There  is  one  and  only  one  parabola  which  passes  through 
three  noncollinear  points  in  the  finite  part  of  a  plane  and  has 
its  infinitely  distant  point  in  a  given  direction. 

This  follows  from  the  theorem  stated  in  §  116,  No.  2. 

^  4.  There  is  one  and  only  one  hyperbola  which  passes  through 
any  point  in  the  finite  part  of  the  plane  and  has  two  given 
straight  lines  as  asymptotes. 

This  follows  from  the  theorem  stated  in  §  116,  No.  3. 

5.  There  is  one  and  only  one  hyperbola  which  has  two  given 
lines  as  asymptotes  and  is  tangent  to  a  third  liiie  which  is  not 
parallel  to  either  of  the  others. 

This  follows  from  the  theorem  stated  in  §  116,  No.  4. 

6.  There  is  one  and  only  one  parabola  ivhich  is  tangent  to 
each  of  three  nonconcurrent  lines  lying  in  the  finite  part  of  the 
plane  and  has  its  infinitely  distant  point  in  a  given  direction. 

This  follows  from  the  theorem  stated  in  §  116,  No.  5. 

4-  7.  There  is  one  and  only  one  parabola  which  is  tangent  to 
any  four  lines  of  a  plane,  no  three  of  which  are  concurrent  and 
no  two  of  which  are  parallel. 

This  follows  from  the  theorem  stated  in  §  116,  No.  6. 


PASCAL'S  THEOREM  147 

Theorem.  Special  Interpretation  of  Pascal's  Theorem 

156.  The  chords  from  a  point  on  a  hyperbola  to  each  of 
Uvo  other  points  on  the  hyperbola  intersect  the  lines  through 
these  two  points  parallel  to  one  of  the  asymptotes,  the  inter- 
sections being  collinear  with  the  intersection  of  the  tangents  at 
the  two  points. 

The  proof  of  this  theorem  is  included  in  the  proof  given  in  §  157. 

Theorem.  Further  interpretation  of  Pascal's  Theorem 

xl57.  The  chords  from  a  point  on  a  parabola  to  each  of 
two  other  points  on  the  parabola  intersect  the  lines  from  these 
tivo  points  to  the  infinitely  distant  point  of  the  curve,  the  inter- 
sections being  collinear  with  the  intersection  of  the  tangents 
at  the  two  points. 

Proof.  These  two  theorems  are  closely  related,  being 
obtained  by  applying  to  such  conies  the  case  of  Pascal's 
theorem  in  which  two  pairs  of  vertices  coincide.  If  a 
conic  is  known  to  have  one  point  at  infinity,  it  may  be 
either  a  hyperbola  or  a  parabola. 

Consider  a  hexagon  inscribed  in  a  conic  in  such  a  way 
that  the  first  and  second  vertices  coincide,  the  fourth  and 
fifth  vertices  coincide,  and  the  sixth  vertex  is  at  infinity. 
We  may  also  assume  that  the  curve  is  a  hyperbola  or  that 
it  is  a  parabola.  If  it  is  a  hyperbola  the  sides  of  the 
hexagon  which  intersect  at  the  infinitely  distant  point  are 
parallel  to  the  same  asymptote.  In  either  case  one  pair 
of  opposite  sides  is  a  pair  of  tangents. 

The  two  theorems  considered  above  are  merely  state- 
ments of  Pascal's  theorem  for  the  two  cases  described, 
the  terms  used  being  appropriate  in  connection  with  these 
two  kinds  of  conies. 


148    CONICS  AND  THE  ELEMENTS  AT  INFINITY 

Theorem.  Special  interpretation  of  Brianchon's  Theorem 

158.  Cfiven  five  tangents  to  a  parahola,  the  line  parallel 
to  the  first  tangent  and  concurrent  with  the  third  and  fourth 
tangents  cuts  the  line  parallel  to  the  fifth  tangent  and  concur- 
rent with  the  second  and  third  tangents  on  the  line  joining  the 
intersection  of  the  first  and  second  tangents  to  that  joining  the 
fourth  and  fifth. 

Proof.  In  Brianchon's  theorem  (§  121),  simply  let  one 
of  the  tangents  be  the  line  at  infinity,  and  the  proof 
follows  at  once. 

Pascal's  theorem  and  Brianchon's  theorem  have  a  large  number 
of  special  interpretations.  Of  these  we  have  space  for  only  the  three 
given  in  §§  156-158.  They  have  been  selected  not  because  of 
their  intrinsic  importance  but  because  they  indicate  the  method 
of  procedure. 

159.  Special  Constructions.  On  account  of  the  elements 
at  infinity  the  problems  which  were  considered  in  Chap- 
ter X  may  also  be  given  special  statements  for  certain 
cases.  Thus  a  point  at  infinity  may  be  specified  by  its 
direction ;  and  since  a  hyperbola  is  determined  by  means 
of  any  three  of  its  points  in  the  finite  part  of  the  plane 
and  its  two  points  at  infinity,  it  is  determined  by  the  three 
points  mentioned  and  the  directions  of  the  two  points  at 
infinity.  These  latter  directions  are  also  the  directions  of 
the  asymptotes. 

In  actual  constructions  certain  special  situations  arise. 
Thus,  drawing  a  line  to  a  given  infinitely  distant  point  is 
the  same  as  drawing  a  line  parallel  to  a  given  line.  To 
effect  this  with  the  ungraduated  ruler  it  is  necessary  to 
have  additional  data,  as  in  the  exercises  on  pages  99  and 
100.  Problems  involving  considerations  of  this  sort  will 
be  considered  in  §§  160-162. 


PROBLEMS  OF  CONSTRUCTION  149 

Problem,  construction  of  the  Hyperbola 

160.  Given  in  a  plane  three  noncoUinear  points  and  tivo 
pairs  of  parallel  lines,  each  pair  having  a  direction  different 
from  those  of  the  lines  joining  the  three  poitits  and  also 
different  from  that  of  the  other  pair,  construct  a  hyperbola 
through  the  three  given  points  and  having  asymptotes  parallel 
to  the  two  given  pairs  of  lines. 

The  student  should  write  out  the  solution,  making  appropriate 
modifications  of  the  methods  employed  in  §§  126  and  138. 

Problem.  Determination  of  a  Conic 

161.  Given  a  set  of  elements  sufficient  for  the  determination 
of  a  conic,  determine  the  nature  of  the  conic  and  the  directions 
of  its  infinitely  distant  points  (if  there  are  any^. 

Among  the  constructions  of  the  second  order  the  construction  in 
§  141  deserves  attention  in  this  connection,  and  this  problem  is  one 
of  its  special  forms.  If  elements  sufficient  for  the  determination  of 
the  conic  are  given,  the  finding  of  the  intersections  of  the  conic 
with  the  line  at  infinity  includes  determining  whether  the  conic  is 
an  ellipse,  a  parabola,  or  a  hyperbola. 

As  in  the  original  case,  if  five  points  of  the  conic  are  not  given 
they  may  be  found  by  construction.  Let  them  be  P^,  P^,  P^,  P^,  Py 
The  triads  of  lines  P^P^,  PJ\,  P^P^  and  P^P^,  P^P^,  P^P^  deter- 
mine the  projectivity  by  means  of  which  any  number  of  additional 
points  of  the  conic  may  be  found. 

A  difficulty  now  arises  in  following  the  original  construction, 
because  the  line  p  is  at  infinity.  The  triads  of  points  of  the  super- 
posed projective  ranges  on  this  line  that  are  determined  by  the  triads 
of  the  flat  pencil  are  now  not  available  from  the  point  of  view  of 
construction  by  the  ruler.  If,  however,  the  possibility  of  drawing 
lines  parallel  to  all  given  lines  is  assumed,  the  resulting  diflSculty 
disappears.  For  the  purpose  of  drawing  the  necessary  parallels,  the 
compasses  must  be  used  more  freely  than  in  the  construction  in 
§  141.  With  this  difference  the  construction  follows  as  before,  and 
the  student  should  write  out  the  solution  in  full. 


160    CONICS  AND  THE  ELEMENTS  AT  INFINITY 

Problem.  Construction  of  the  Parabola 

162.  Given  four  points  in  a  plane,  no  three  of  them  col- 
inear,  construct  the  parabola  which  passes  through  these  points. 

By  §  144  the  number  of  such  parabolas  is  none,  one,  or  two. 
The  solution  of  this  problem,  which  is  based  on  that  of  §  143,  is 
left  for  the  student. 

Exercise  30.   Elements  at  Infinity 

As  suggested  on  pages  144-148,  investigate  with  respect  to 
the  elements  at  infinity  the  following  cases  already  considered : 


1. 

§120. 

5. 

§128. 

9. 

Page  142,  Ex.  19. 

2. 

§121. 

6. 

§149. 

10. 

Page  142,  Ex.  22. 

3. 

§126. 

7. 

Page  141,  Ex.  10. 

11. 

Page  142,  Ex.  28. 

4. 

§127. 

8. 

§147. 

12. 

Page  142,  Ex.  29. 

In  Exs.  1-12  practice  in  special  interpretation,  not  the  finding  of  im- 
portant results,  is  the  object. 

13.  Using  tlie  compasses  only  once  arid  the  ruler,  find  five 
points  in  the  finite  part  of  the  plane  which  shall  determine  an 
ellipse  other  than  a  circle. 

14.  Consider  Ex.  13  for  the  case  of  a  parabola. 

15.  Consider  Ex.  13  for  the  case  of  a  hyperbola. 

16.  In  the  finite  part  of  the  plane  find  four  points  through 
which  no  parabola  passes. 

17.  In  the  finite  part  of  the  plane  find  four  points  through 
which  two  parabolas  pass. 

18.  Find  four  points  through  which  both  a  circle  and  a 
parabola  pass. 

19.  On  a  straight  line  p  passing  through  a  given  point  Pj, 
find  a  point  P  such  that  through  P^,  P  and  two  other  given 
points  P^y  Pg  there  pass  (1)  two  parabolas ;  (2)  one  parabola ; 
(3)  no  parabola.  In  each  case  indicate  all  possible  positions  of  P. 


CHAPTER  XII 


POLES  AND  POLARS  OF  COXICS 

163.  Polar  of  a  Point.  In  §  165  it  will  be  proved  that 
if  three  lines,  concurrent  at  a  point  0,  cut  a  conic  in 
A^,  A^ ;  B^,  B^ ;  Cj,  Cg,  the  harmonic  conjugates  Ay  B^  C 
of  0  with  respect  to  A^,  A^; 
J?j,  B^;  Cj,  Cj  are  collinear. 
Since  OC^C^  may  be  any  line 
through  0,  it  follows  that  all 
harmonic  conjugates  of  0  with 
respect  to  the  pairs  of  points 
in  which  Imes  through  0  cut 
the  conic  are  on  the  line  de- 
termined by  A  and  B,  two  of 
these  conjugates. 

The  line  thus  determined  by  two  harmonic  conjugates 
is  called  the  polar  of  the  point  0  with  respect  to  the  conic. 

164.  Pole  of  a  Line.  Suppose  that  there  are  given  a  conic 
and  a  line  o.  If  from  a  point  of  the  Ime  o  two  tangents 
are  drawn  to  the  conic,  then 
ky  the  harmonic  conjugate  of 
o  with  respect  to  the  two  tan- 
gents, may  be  constructed. 
This  line  will  be  spoken  of  as 
a  harmonic  conjugate  of  o  with 
respect  to  the  conic,  or  simply 
as  a  harmonic  conjugate  of  o.  The  point  of  intersection  of 
two  harmonic  conjugates  of  o  will  be  called  the  pole  of  o. 

151 


152 


POLES  AND  POLARS  OF  CONICS 


Theorem.  Triangles  m  Homology 

165.  If  through  a  point  0  three  lines  are  drawn  cutting 
a  conic  in  the  pairs  of  points  A^,  A^ ;  B^,  B^ ;  Cj,  Cg,  the 
triangles  AyB^C-^  «wt?  -42^2^2  ^^^  *^^  harmonic  homology. 


Proof.  The  triangles  A^B^  Cj  and  A^B^C^  are  in  homology, 
as  is  indicated  in  the  note  under  Ex.  13,  page  13. 

The  axis  of  homology  passes  through  X^,  Xj,  A'g,  the 
intersections  of  B^C^,  B^C^\   C^Ay,  C^A^\  A^B^,  A^B^. 

The  Pascal  line  of  the  hexagon  A^C^C^A^B^B^  contains 
the  points  L,  0,  M,  which  are  the  intersections  of  the  oppo- 
site sides  ^jCj,  A^B^;  C^C^,  B^B^;  C^A^,  A^By  Moreover, 
the  Pascal  line  cuts  the  line  AjAg  in  a  point  N. 

Also  range  LMON  =  range  A^A^  OA 

—  range  OAA^A^ 
But  (^LM0N)  =  -1. 

Therefore  (OAA^A^^  =  1  ^  (OAA^A^)  =  - 1. 

Hence  the  constant  of  homology  is  —1. 


§23 

§30 
§24 


POLES  AND  POLARS  153 

Exercise  31.    Poles  and  Polars 
Prove  the  theorem  of  §  165  for  the  following  triangles  : 
1.  A^B^C^,  A^B^C^.    2.  A^B^C^,  A^B^C^.    3.  A^B^C^,  A^B^C^. 

4.  In  the  figure  of  §165  A^B^,  A^B^;  A^C^,  A^C^,;  Bf^ 
B^C^  also  intersect  on  the  line  X^X^X^. 

5.  Draw  a  large  and  accurate  figure  consisting  of  the  figure 
of  §  165  and  the  additional  lines  which  would  be  introduced 
in  proving  Exs.  1,  2,  and  3. 

It  is  suggested  that  the  sets  of  lines  introduced  on  account  of  Exs.  1-3 
be  given  distinctive  colors. 

6.  State  and  prove  the  dual  of  §  165  for  the  plane. 

7.  Construct  carefully  the  figure  which  is  the  dual  of  that 
required  in  Ex.  5. 

8.  If  two  triangles  are  homologic  and  the  constant  of 
homology  is  —1,  the  six  vertices  are  on  a  conic. 

9.  If  two  triangles  are  homologic  and  the  constant  of 
homology  is  —  1,  the  six  sides  are  tangent  to  a  conic. 

10.  By  means  of  §  165  find  a  figure  harmonically  homo- 
logic  with  any  polygon  inscribed  in  a  conic. 

11.  Inscril)e  in  a  conic  a  polygon  which  shall  be  harmoni- 
cally self-homologic. 

12.  Use  §  165  to  obtain  a  line  that  bisects  a  given  set  of 
parallel  chords  of  a  conic. 

13.  Given  a  circle  or  a  carefully  drawn  ellipse,  parabola,  or 
hyperbola,  show  experimentally  that  the  polar  of  a  given 
point  O  is  the  same  line  whatever  pair  of  three  given  lines 
through  O  is  used  in  constructing  it. 

14.  Construct  the  polar  o  of  a  given  point  O  of  a  conic,  and 
then  find  the  intersection  of  the  polars  of  two  points  on  o. 

15.  Construct  the  polar  of  a  vertex  of  the  diagonal  triangle 
of  a  complete  quadrangle  inscribed  in  a  conic. 


154  POLES  AND  POLARS  OF  CONICS 

166.  Properties  of  a  Polar.  The  polar  of  a  point  with 
respect  to  a  conic  has  the  following  important  properties: 

1.  The  polar  of  a  point  0  with  respect  to  a  conic  con- 
tains all  harmonic  conjugates  of  0  with  respect  to  the  conic. 

In  the  preceding  discussion  the  polar,  though  deter- 
mined by  A  and  B  only,  contains  C,  no  matter  what  is  the 
direction  of  OC^C^.  Accordingly,  any  two  of  the  harmonic 
conjugates  of  O  determine  a  line  through  all  of  them. 

In  each  of  these  cases  the  student  should  draw  the  figure  and  be 
certain  that  the  suggested  proof  is  clearly  followed. 

2.  The  polar  of  a  point  O  with  respect  to  a  conic  contains 
the  other  intersections  of  opposite  sides  of  any  inscribed  com- 
plete quadrangle  of  which  0  is  a  diagonal  point. 

In  the  discussion  of  §  165,  A^B^A^B^  is  any  inscribed 
quadrangle  of  which  O  is  a  diagonal  point;  and  the  other 
intersections  of  pairs  of  opposite  sides  of  this  quadrangle 
are  situated  on  the  axis  of  homology,  which  coincides  with 
the  polar  of  0. 

3.  The  polar  of  a  point  0  with  respect  to  a  conic  contains 
the  intersections  of  pairs  of  tangents  to  the  conic  at  the  points 
in  which  any  line  through  0  cuts  the  conic. 

Let  the  line  OB^B^  (§165)  approach  coincidence  with 
the  line  OA^A^.  Then  B^A^  approaches  the  tangent  at  Jj* 
and  B^A^  approaches  the  tangent  at  A^\  and  at  all  stages 
these  lines  intersect  on  the  polar  of  0. 

4.  The  polar  of  a  point  0  with  respect  to  a  conic  contains  \ 
the  points  of  contact  of  the  tangents  (if  there  are  any^  from 
O  to  the  conic. 

If  the  line  OA^A^  rotates  about  0  and  approaches  the 
position  of  tangency  to  the  conic,  the  points  A^^  A,  A^ 
approach  coincidence  at  the  point  of  contact. 


PROPERTIES  OF  POLES  AND  POLARS       155 

167.  Properties  of  a  Pole.  Applying  the  Principle  of 
Duality  to  the  statements  of  §  166,  we  have  the  following  ; 

1.  The  pole  of  a  line  o  with  respect  to  a  conic  is  on  all 
harmonic  conjugates  of  o  with  respect  to  the  conic. 

The  student  should  draw  the  figure  in  each  of  these  cases  and 
should  write  out  the  duals  of  the  proofs  suggested  in  §  166. 

2.  The  pole  of  a  line  o  with  respect  to  a  conic  is  on  the 
other  lines  joining  opposite  vertices  of  any  circumscribed  com- 
plete quadrilateral  of  which  o  is  a  diagonal  line, 

3.  The  pole  of  a  line  o  vrith  respect  to  a  conic  is  on  the 
chord  of  contact  (^produced  if  necessary^  of  the  tangents  from 
any  point  of  o  to  the  conic. 

4.  The  pole  of  a  line  o  with  respect  to  a  conic  is  the  in- 
tersection of  the  tangents  to  the  conic  at  the  points  (if  there 
are  any^  in  which  the  line  o  cuts  the  conic. 

A  comparison  of  the  properties  of  pole  and  polar  as  stated  in 
§§  166  and  167  leads  to  various  interesting  conclusions.  A  few  of 
these  are  stated  in  §§  168-171. 

Exercise  32.    Construction  of  Poles  and  Polars 

1.  Give  a  construction  based  upon  §  166,  2,  for  the  polar  of 
a  given  point  with  respect  to  a  given  conic. 

2.  Construct  the  tangents  to  a  conic  from  a  given  point  O. 

3.  Find  the  pole  of  a  given  line  with  respect  to  a  conic, 

4.  At  a  given  point  on  a  conic  draw  a  tangent  to  the  conic. 

5.  For  any  conic  construct  the  polar  o  of  a  given  point  0, 
and  then  find  the  pole  of  the  line  o. 

The  figure  should  be  drawn  very  carefully. 

6.  With  respect  to  a  given  conic  find  the  polar  o^  of  a  given 
point  Oj,  the  polar  o^  of  a  given  point  O^  on  o^,  and  the  polar  o^ 
of  the  intersection  of  o^  and  o^. 


156  POLES  AND  POLARS  OF  CONICS 

Theorem.  Relation  of  Pole  and  Polar 

168.  If  a  'point  0  is  the  pole  of  the  line  o,  the  line  o  is  the 
polar  of  the  point  0. 


O 


Proof.  Let  0  be  the  pole  of  the  line  o,  and  let  Op  0^ 
be  pomts  on  o.  Let  the  chords  of  contact  of  the  tangents 
to  the  curve  from  Oj  and  O^  cut  o  in  Og  and  0^.  These 
chords  pass  through  0  (§  167,  3).     Draw  0^0  and  O^O. 

Since  the  lines  0^0,  0^0  are  conjugates  of  the  line  o 
(§  167,  1),  the  points  Og,  O^  are  conjugates  of  0,  and 
therefore  o  is  the  polar  of  0  (§  164). 

169.  Inside  and  Outside  of  a  Conic.  If  a  point  0  moves 
up  to  a  position  on  a  conic,  its  polar  o  becomes  a  tangent ; 
but  if  O  is  not  on  the  conic,  either  no  tangent  or  two  tan- 
gents pass  through  it.  According  as  0  is  on  two  tangents 
or  on  no  tangent,  it  is  said  to  be  outside  or  inside  the  curve. 

If  0  is  outside  the  curve,  its  polar  cuts  the  conic  in  the 
two  points  of  contact  of  the  tangents  from  0  to  the  conic. 

Suppose  O  is  inside  the  curve  ;  then,  smce  the  polar  meets 
all  tangents  to  the  conic,  infinitely  many  of  its  points  are 
outside  the  curve.  Moreover,  the  polar  does  not  meet  the 
curve ;  for  if  it  did,  tangents  could  be  drawn  from  0  to  the 
intersections.  The  proof  of  the  theorem  that  every  point 
of  the  polar,  that  is,  every  harmonic  conjugate  of  0,  is 
outside  the  conic  is,  however,  somewhat  complicated  and 
will  be  omitted  from  this  book,  the  fact  being  assumed. 


RELATIONS  OF  POLES  AND  POLARS         157 

Theorem,  points  on  a  polar 

170.  If  the  'point  O^  is  on  the  polar  of  the  point  Og,  then 
Oj  is  on  the  polar  of  the  point  Oj. 


Proof.  Of  the  points  Oj  and  0^,  at  least  one  must  be 
outside  the  conic.  For  if  Oj  and  0^  are  both  inside  tlie 
conic,  then  (§  169)  every  point  of  o^,  including  Oj,  is  out- 
side the  conic,  which  is  contrary  to  the  hypothesis  made. 

Let  O2  be  outside  the  conic.  Then  §  166,  3  and  4,  yields 
the  desired  conclusion. 

Let  O2  be  inside  the  conic.  Then  Oj,  being  on  Og,  is  out- 
side the  conic.  Let  Oj  cut  the  conic  in  the  points  A,  B. 
It  will  cut  the  line  Oj  0^  in  0<^ ;  for  otherwise  the  tangents 
of  which  O^A  is  the  chord  of  contact  would  not  meet  on 
©2,  as  they  must  by  §  166,  3.    Hence  O3  is  on  0^. 

171.  Corollary.  If  a  point  0^  traces  out  a  range  whose 
base  is  o^,  its  polar  Oj  traces  out  a  fat  pencil  whose  base 
is   Og,  the  pole  of  o^. 

The  relation  between  the  range  traced  by  the  point  O^  and  the 
flat  pencil  described  by  o^  is  stated  in  the  theorem  of  §  177. 

172.  Conjugate  Points  and  Conjugate  Lines.  Two  points 
so  situated  that  each  is  on  the  polar  of  the  other  are  said 
to  be  conjugate^  and  two  lines  so  situated  that  each  contains 
the  pole  of  the  other  are  said  to  be  conjugate. 

Accordingly,  harmonic  conjugates  are  sjiecial  cases  of  conjugates. 


158         POLES  a:n^d  polars  of  conics 

173.  Self-Polar,  or  Self-Conjugate,  Triangle.  If  any  point 
Oj,  not  on  a  conic,  is  taken,  its  polar  Oj  may  be  found.  On 
this  line  let  any  point  0^,  not  on  the  conic,  be  taken,  and 
let  its  polar  o^  be  found.  Then  o^  passes  through  Oj.  Let 
the  intersection  of  Oj  and  o^  be  Og. 

The  polar  Og  of  Og  is  the  line  0^0^,  and  Og  is  conjugate 
to  both  Oj  and  Oj.  The  triangle  O^O^O^  is  such  that  each 
side  is  the  polar  of  the  op- 
posite vertex.  Every  such 
triangle  is  said  to  be  self- 
polar,  or  self-conjugate,  with 
respect  to  the  conic.  Evi- 
dently there  is  an  infi- 
nite number  of  triangles 
which   are    self-polar   with   respect   to    a   given    conic. 

No  self -polar  triangle  has  two  vertices  inside  the  conic ; 
for  if  one  vertex  is  inside  the  curve,  its  polar  in  which  the 
other  two  vertices  lie  is  entirely  outside  the  curve. 

If  we  should  attempt  to  construct  a  self-polar  triangle 
all  of  whose  vertices  are  outside  the  conic,  we  might 
choose  a  point  Oj  outside  the  conic  and  draw  its  polar  Oj. 
Let  A,  B  be  the  two  points  in  which  this  line  would  cut 
the  conic.  Then  the  other  vertices  0^,  Og  would  be  on  Oj 
and  would  be  separated  by  A,  B.  If  we  should  take  0^ 
outside  the  conic,  it  would  remain  to  determine  whether  0^ 
would  be  inside  or  outside  the  conic;  that  is,  whether 
tangents  could  be  drawn  from  Og  to  the  conic.  The  con- 
siderations adduced  thus  far  would  not  enable  us  to  give 
a  sufficiently  brief  but  complete  discussion  of  this  question, 
but  an  application  of  principles  of  continuity,  which  have 
'not  been  developed  in  this  book,  would  enable  us  to  go 
farther  and  to  establish  the  proposition  that  every  self- 
polar  triangle  has  one  and  only  one  vertex  within  the  conic 


SELF-POLAR  TRLA.NGLE 


159 


Theorem.  Diagonal  Triangle 

174.  The  diagonal  triangle  of  a  complete  quadrangle 
inscribed  in  a  conic  is  self -polar ;  and,  conversely,  a  self-polar 
triangle  is  the  diagonal  triangle  of  an  inscribed  complete 
quadrangle. 


Proof.  Let  O^O^O^  (Fig- 1)  be  the  diagonal  triangle  of  a 
complete  quadrangle  A^A^B^B^  inscribed  in  a  given  conic. 
From  §  166,  2,  it  follows  that  O^O^O^  is  a  self-polar  triangle. 

Conversely,  let  OjOgOg  (Fig.  2)  be  a  self-polar  triangle 
with  respect  to  a  given  conic. 

From  A^j  any  point  on  the  conic,  draw  A^O^,  A^O^,  and 
let  them  meet  the  conic  again  in  A^,  B^.  Draw  O^B^, 
O^B^,  O^A^,  O^A^,  and  let  O^A^,  O^A^  meet  in  B^.  Also 
let  Og^j,  OjOg  meet  in  K,  let  O^A^,  ^3^1  meet  in  L,  and 
let  OjJj,  OgOg  meet  in  M. 

Then  (^A^B^KO^^  =  -l  =  {A^A^O^Ly 

Hence        r&n^e  A^B^KO^^TdmgQ  A^A^O^L, 
and  the  points  A^,  B^,  Oj  are  collinear. 

Again,  since  range  A^B^MO^  =  harmonic  range  A^A^O^L 
and  M  is  on  the  polar  of  Oj,  then  B^  is  on  the  conic. 
Hence  the  quadrangle  A^A^B^B^  is  inscribed. 


160  POLES  AKD  POLARS  OF  CONICS 

Theorem.  Ranges  and  their  Conjugates 

175.   On  a  line  the  range  of  points  and  the  range  of  their 
conjugates  with  respect  to  a  conic  constitute  an  involution. 


Proof.  Let  0^  be  the  pole  of  a  line  o^  with  respect  to 
a  given  conic.  Through  0^  draw  lines  cutting  the  conic 
in  ^1,  J  J ;  and  also  in  J3^,  B^  ;  Bl,  B^  ;  B[',  B^';  ....  Then 
OjOjOg  is  the  diagonal  triangle  of  Aj^A^B^B^;  0^0{0^  of 
A^A^B'^B[ ;  •  .  ..  The  pairs  of  points  0^,  0^\  0[,  0'^',  --  - 
are  conjugate,  and,  associating  with  A^A.^  all  lines  through 
Oj,  we  obtain  all  pairs  of  conjugates  on  o^. 

But        range  0^0[0i;  . .  .  OgO^O^'  • .  • 

=  pencil  A,(iO,0[0!^  .  •  •  O^O'^O'^  • .  •) 
^  pencil  ACOjOi'Oi"  . .  •  0,0',0'^  . . .) 
^  pencil  A^^B^B'^B'^  . . .  B^B[B!<  . . .) 
-^  pencil  A^  {B^B'^B'^  •  •  •  B^B[B'^  • .  •) 
grange  030^0^'...0i0{0i".... 


ELEMENTS  AND  THEIR  CONJUGATES        161 

Theorem,  pencils  and  their  Conjugates 

176.  The  pencil  of  lines  through  a  point  and  the  pencil  of 
their  conjugates  with  respect  to  a  conic  constitute  an  involution. 

The  proof  of  this  dual  of  §  175  is  left  for  the  student. 

The  properties  of  poles  and  polars  furnish  one  basis  for  the  estab- 
lishment of  the  validity  of  the  Principle  of  Duality  for  figures  in 
a  plane,  and  Poncelet  practically  used  them  in  this  way. 


Theorem,  point  Describing  a  Range 

177.  If  a  point  describes  a  range,  the  polar  of  the  point 
with  respect  to  a  conic  describes  aflat  pencil  which  is  projective 
with  that  range. 

O 


Proof.  Let  O  be  the  pole  of  the  range,  and  let  the  point 
take  the  positions  Oj,  Og,  Og,  O^,  •  •  ..  Then  its  polar 
always  passes  through  O.  The  polars  of  Oj,  Og,  Og,  O4  •  •  • 
intersect  0,  the  base  of  the  range,  in  0[,  0^,  O3,  0^,  •  •  •,  the 
conjugates  of  Oj,  O^,  Og,  O4,  •  •  •  respectively. 

Then      range  O^O^O^O^  .  •  •  ^  range  0[0'^0^0[  ...  §  175 

=  pencil  OjOjOgO^  .... 

Hence     range  O^O^O^O^- '  --r  pencil  o-^o^o^o^  .... 


162  POLES  AND  POLARS  OF  CONICS 

178.  Duality  in  Plane  Figures.  It  is  now  possible  to  indi- 
cate a  line  of  argument  by  which  the  Principle  of  Duality 
may  be  established  for  plane  geometry.  In  the  plane  of  a 
given  figure,  composed  of,  or  generated  by,  points  and  lines, 
take  any  conic  and  construct  the  polar  of  every  point  and 
the  pole  of  every  line.  Then  a  new  figure  is  obtained  in 
which  there  is  a  point  for  every  line  and  a  line  for  every 
point  of  the  first  figure,  and  in  which  to  the  intersection  of 
any  two  lines  of  the  first  figure  there  corresponds  the  line 
determined  by  the  poles  of  these  two  lines  in  the  second. 
To  any  locus  of  points  in  the  first  figure  there  corresponds 
an  envelope  of  lines  in  the  second.  Hence  it  is  evident  that 
a  duality  in  figures  exists. 

179.  Duality  in  Properties  of  Figures.  Likewise,  if  any 
nonmetric  proposition  is  true  for  some  or  all  of  the  points 
and  lines  of  the  first  figure  in  §  178,  it  follows  that  this  figure 
cannot  be  constructed  by  choosing  arbitrarily  in  the  plane  the 
sets  of  points  and  lines  which  constitute  it,  but  that,  certain 
points  and  lines  being  selected,  the  choice  of  the  remaining 
ones  is  restricted.  For  the  proposition,  by  its  assertion  of  a 
relation,  or  of  relations,  existing  among  the  points  and  the 
lines  of  a  given  figure,  is  a  denial  of  the  possibility  of  choos- 
ing all  of  them  arbitrarily.  Hence  the  second  figure  is  not 
merely  a  set  of  lines  and  points,  each  chosen  arbitrarily  in  the 
plane.  In  fact,  there  exists  a  certain  limitation  upon  the 
choice  of  the  lines  and  points  of  the  second  figure,  and 
the  statement  of  this  limitation  constitutes  the  proposition 
correlative  to  the  one  regarding  the  first  figure.  Hence 
there  is  a  duality  in  properties  of  figures. 

180.  Polar  Reciprocal  or  Polar  Dual.  A  figure  obtained 
from  a  given  figure  by  the  method  explained  in  §  178  is 
called  the  polar  reciprocal  or  polar  dual  of  the  given  figure. 


CENTER  AND  DIAMETERS  OF  A  CONIC      163 

181.  Center  and  Diameters  of  a  Conic.  As  in  some  pre- 
ceding cases,  useful  metric  relations  are  obtained  by  con- 
sideration of  the  elements  at  infinity.  Thus,  the  line  at 
infinity  has  a  pole,  and  from  the  property  of  the  harmonic 
range  this  pole  is  seen  to  bisect  every  chord  of  a  conic 
which  passes  through  it.  Because  of  this  symmetry  of  the 
curve  with  respect  to  the  pole  of  the  line  at  infinity,  this 
point  is  called  the  center  of  the  conic.  If  the  conic  is 
a  parabola,  the  center  is  also  the  point  of  contact  of  the 
parabola  with  the  line  at  infinity;  and  since  this  point  is 
at  infinity  and  the  notion  of  symmetry  loses  its  usual  force, 
the  parabola  is  generally  said  not  to  have  a  center.  In  the 
case  of  the  other  conies  the  center  is  not  at  mfinity,  the 
center  of  the  ellipse  being  inside  the  curve  and  that  of 
the  hyperbola  being  outside.  In  the  case  of  the  latter  it 
is  the  intersection  of  two  tangents  to  the  curve  whose 
points  of  contact  are  at  infinity.  These  tangents  are,  of 
course,  the  asymptotes. 

Again,  every  point  on  the  line  at  infinity  has  a  polar 
which  passes  through  the  center.  The  polar  of  a  point  at 
infinity  is  called  a  diameter  of  a  conic.  In  the  case  of  a 
parabola,  since  all  the  points  at  infinity  are  on  the  line  at 
infinity,  the  diameters  intersect  in  a  point  at  infinity  and 
hence  are  parallel.  Also,  any  point  on  the  line  at  infinity 
being  chosen,  all  lines  through  that  point  are  parallel ;  and 
if  any  one  of  these  lines  meets  the  conic,  the  harmonic  con- 
jugate of  the  chosen  point  at  infinity  that  is  situated  on 
that  line  bisects  the  segment  of  it  which  is  intercepted  by 
the  curve.  Hence,  every  diameter  bisects  each  of  the  set  of 
parallel  chords  of  the  conic  which  passes  through  its  pole. 
Likewise,  the  tangents  at  the  points  in  which  a  diameter 
cuts  the  conic  pass  through  the  pole  of  that  diameter ;  that 
is,  they  are  parallel  to  the  chords  bisected  by  the  diameter. 


164  POLES  AND  POLARS  OF  CONICS 

182.  Conjugate  Diameters  and  Principal  Axes.  If  two 
diameters  are  conjugate  lines  with  respect  to  a  conic,  each 
diameter  passes  through  the  pole  of  the  other,  which  is 
the  point  at  infinity,  and  so  each  is  parallel  to  the  chords 
bisected  by  the  other.  Such  diameters  are  called  conjugate 
diameters.  According  to  §  176  they  constitute  an  involu- 
tion, and  since,  in  general,  only  one  pair  of  corresponding 
lines  of  an  involution  is  at  right  angles,  in  general  only 
one  pair  of  conjugate  diameters  is  at  right  angles.  These 
two  diameters  are  called  the  principal  diameters  and  form 
the  principal  axes.  Moreover,  it  follows  from  §  74  that  if 
there  are  more  pairs  of  conjugate  diameters  which  are  at 
right  angles,  all  pairs  have  this  property.  In  this  case  it 
can  be  shown  that  the  conic  is  a  circle. 

When  the  involution  of  diameters  is  hyperbolic,  the 
self -corresponding  elements  are  the  asymptotes ;  and  these 
separate  harmonically  every  pair  of  conjugate  diameters. 

Two  conjugate  diameters  and  the  line  at  infinity  consti- 
tute a  seK-polar  triangle,  and  of  such  a  triangle  just  two 
sides  cut  the  conic.  Hence  both  of  two  conjugate  diam- 
eters of  an  ellipse  meet  the  curve,  but  only  one  of  any 
two  conjugate  diameters  cuts  the  hyperbola. 

Exercise  33.    Review 

1.  Find  the  locus  of  the  harmonic  conjugates  of  a  given 
point  with  respect  to  a  given  pair  of  straight  lines. 

2.  Find  the  point  which  is  the  harmonic  conjugate  of  a 
given  point  with  respect  to  each  of  two  given  pairs  of 
straight  lines. 

3.  Find  the  point  which  has  an  infinite  number  of  harmonic 
conjugates  with  respect  to  each  of  two  given  pairs  of  straight 
lines,  and  find  the  locus  of  these  conjugates. 


REVIEW  EXERCISES  165 

4.  Find  three  points  each  of  which  is  conjugate  to  the 
other  two  with  respect  to  a  pair  of  opposite  sides  of  a  given 
complete  quadrangle. 

5.  Each  of  the  three  points  found  in  Ex.  4  is  conjugate  to 
the  other  two  with  respect  to  any  conic  which  passes  through 
the  four  vertices  of  the  quadrangle. 

6.  Given  any  five  points,  no  four  of  which  are  collinear, 
construct,  with  the  ruler  only,  the  polar  of  a  given  point  with 
respect  to  that  conic. 

7.  Solve  the  dual  of  Ex.  6. 

8.  Construct  a  self-polar  triangle  for  a  given  conic,  using 
the  ruler  only. 

9.  Construct  the  common  self-polar  triangle  for  all  conies 
which  pass  through  four  given  points. 

10.  If  all  parts  of  a  figure  which  consists  of  a  conic  and 
a  self-polar  triangle  are  erased  except  the  triangle  and  two 
points  of  the  curve,  reconstruct  the  figure. 

11.  Through  three  given  points  construct  a  conic  which  has 
a  given  point  and  a  given  line  as  pole  and  polar. 

12.  Solve  the  dual  of  Ex.  11. 

13.  Through  two  given  points  construct  a  conic  which  is 
tangent  to  a  given  line  and  has  a  given  point  and  a  given  line 
as  pole  and  polar. 

14.  Through  four  given  points  construct  a  conic  which  has 
a  given  pair  of  points  as  conjugates. 

15.  Through  four  given  points  and  through  some  pair  (not 
specified)  of  points  of  a  given  involution  on  a  straight  line 
construct  a  conic. 

16.  Given  four  points  P^,  P^,  P^,  P^  and  two  fixed  lines  j)^, 
p^  passing  through  P^  and  P^  respectively,  find  the  envelope  of 
the  line  joining  the  other  intersections  of  these  two  lines  with 
a  variable  conic  through  the  four  points. 


166  POLES  AND  POLARS  OF  CON^ICS 

17.  Given  the  two  points  common  to  two  conies  and  three 
other  points  of  each,  find  all  the  intersections  of  the  conies. 

18.  Given  five  points  of  a  conic,  find  any  diameter  and  the 
diameter  conjugate  thereto. 

19.  Given  five  points  on  a  conic,  construct  the  center,  the 
axis,  and  the  asymptotes  of  the  conic. 

The  student  should  consult  §  70  and  §  74. 

20.  If  a  conic  has  more  than  one  pair  of  conjugate  diam- 
eters which  are  at  right  angles,  the  conic  is  a  circle. 

21.  Given  six  points  on  a  conic  and  the  tangents  at  these 
points,  the  Pascal  line  of  the  inscribed  hexagon  is  the  polar 
of  the  Brianchon  point  of  the  circumscribed  hexagon. 

22.  If  the  pole  of  the  line  PJ^^  with  respect  to  one  conic 
which  passes  through  the  points  P^,  P^,  P^,  P^  coincides  with 
the  pole  of  P^P^  with  respect  to  a  second  conic  through  these 
points,  the  pole  of  -P^Pg  with  respect  to  the  second  conic  coin- 
cides with  the  pole  of  ^8^4  with  respect  to  the  first. 

23.  Construct  a  conic  which  has  a  given  triangle  as  a  self- 
polar  triangle  and  a  given  point  and  a  given  line  as  pole 
and  polar. 

24.  Construct  a  conic  which  has  a  given  point  as  center 
and  a  given  self-polar  triangle. 

25.  Construct  a  conic  which  has  a  given  pair  of  lines  as 
conjugate  diameters  and  a  given  point  and  a  given  line  as  pole 
and  polar. 

26.  Construct  a  conic  that  has  each  side  of  a  given  pentagon 
as  polar  of  the  vertex  opposite  to  it. 

27.  In  the  construction  of  Ex.  26  find  the  polar  reciprocal  of 
the  conic  determined  by  the  vertices  of  the  pentagon. 

28.  If  a  moving  point  traces  a  given  conic,  find  the  envelope 
of  the  polar  of  the  point  with  respect  to  another  given  conic. 

29.  The  lines  joining  the  vertices  of  a  triangle  to  the  corre- 
sponding vertices  of  the  triangle  polar  to  it  are  concurrent. 


CHAPTER  XIII 
QUADRIC  CONES 

183.  Properties  of  Quadric  Cones.  A  large  number  of  the 
properties  of  quadric  cones  can  be  derived  as  duals  of 
those  of  the  conic.  Thus,  it  is  evident  that  a  quadric 
cone  is  determined  by  its  relation  to  certain  sets  of  five 
planes  and  lines.  Moreover,  the  theorems  of  Pascal, 
Brianchon,  and  Desargues,  and  their  limiting  cases,  have 
duals  which  relate  to  the  tangent  planes  and  generating 
lines  of  the  quadric  cones. 

In  the  problems  of  construction  of  the  first  order  the 
possibility  of  drawing  lines  through  pairs  of  points  and  of 
finding  pomts  as  the  intersections  of  lines  was  assumed. 
For  the  corresponding  problems  of  this  chapter  there 
should  be  assumed  the  possibility  of  drawing  lines  common 
to  pairs  of  planes  and  of  constructing  planes  determined 
by  pairs  of  lines. 

In  the  problems  of  the  second  order  the  constructibility 
of  at  least  one  conic,  ordinarily  a  circle,  was  assumed.  At 
this  point  the  corresponding  assumption  is  that  of  the  con- 
structibility of  at  least  one  quadric  cone.  On  the  basis 
of  these  assumptions  the  problems  analogous  to  those  of 
Chapter  X  can  be  adequately  treated. 

Similarly,  the  theory  of  polar  planes  and  lines  of  quad- 
ric cones  follows  from  that  of  poles  and  polars  of  conies, 
and  can  be  made  to  furnish  an  evidence  of  the  truth  of 
the  Principle  of  Duality  for  the  bundle..  The  development 
of  the  subject  along  these  lines  is  left  as  an  exercise. 

167 


168  QUADRIC  CONES 

Theorem.  Sections  of  Quadric  Cones 

184.  A  quadric  cone  the  vertex  of  which  is  not  infinitely 
distant  may  he  so  cut  by  a  plane  as  to  yield  an  ellipse,  a 
parabola,  or  a  hyperbola. 


Fig.  1 

Proof.  Every  quadric  cone  can  be  generated  by  means 
of  the  lines  of  intersection  of  corresponding  planes  of 
projective  axial  pencils  that  belong  in  the  same  bundle, 
and  every  plane  section  of  a  quadric  cone  is  a  conic. 

Suppose  now  that  a  quadric  cone  (Fig.  1)  is  generated 
from  the  intersections  a^,  a^,  a^,  '  -  •  of  pairs  of  correspond- 
ing planes  a^,  a^ ;  a^,  a^;  a^,  a^;  •  •  -  of  two  axial  pencils 
whose  bases  are  the  lines  p  and  p' ;  and  let  the  base  of  the 
bundle  be  P,  a  point  not  at  mfinity. 

First,  to  secure  a  plane  section  which  is  a  hyperbola,  let 
TT  be  a  plane  not  through  P  but  parallel  to  a^  and  a^. 

This  plane  cuts  the  cone  in  a  conic,  and  since  the  plane 
cuts  aj  and  a^  at  infinity,  the  conic  has  two  distinct  points 
at  infinity.  The  conic  is  not  a  pair  of  straight  lines,  since 
its  projector  from  P  is  not  a  pair  of  planes. 

Hence  the  conic  is  a  hyperbola. 


SECTIONS  OF  QUADRIC  CONES 


169 


Next,  to  obtain  a  section  which  is  a  parabola,  let  a[  be  the 
plane  tangent  to  the  cone  (Fig.  2)  along  the  generator  a^. 

This  plane  contains  the  whole  of  the  line  a^  but  meets 
any  other  generator,  as  a^,  in  only  one  point,  namely,  P. 

Cut  the  cone  by  a  plane  a^  parallel  to  a[.  This  plane 
cuts  any  generator  other  than  a^,  as  a^^  at  a  finite  distance 
from  P,  and  it  cuts  aj  at  infinity. 

Hence  the  section  of  the  cone  has  one  and  only  one 
point  at  mfinity,  and  the  conic  is  a  parabola. 


Fig.  2 


Fig.  3 


Finally,  to  obtain  a  section  which  is  an  ellipse,  let  ttj  be 
a  plane  (Fig.  3),  through  P  but  not  coincident  with  any 
plane  of  either  axial  pencil,  and  let  ttj  be  a  plane  not 
through  P  but  parallel  to  tt^  at  a  finite  distance  from  it. 

The  intersection  of  any  pair  of  corresponding  planes,  as 
a  and  a',  since  it  meets  ir^  at  P,  cannot  be  parallel  to  7r2. 

Hence  the  intersection  of  ttj  and  the  cone  has  no  point 
at  infinity,  and  the  conic  is  an  ellipse. 

If  the  vertex  of  the  cone  is  at  infinity,  the  generating  lines  are 
all  parallel.  It  will  be  seen  (§  186)  that  the  surface  must  contain  no, 
one,  or  two  infinitely  distant  generators.  In  these  cases  the  sections 
made  by  planes  not  parallel  to  the  generators  will  be  all  ellipses,  all 
parabolas,  or  all  hyperbolas  respectively.    Hence  the  theorem  fails. 


170  QUADRIC  CONES 

185.  Axes  of  a  Quadric  Cone.  Let  o  be  a  line  through 
the  vertex  of  a  quadric  cone,  and  let  a>  be  the  polar 
plane  of  o  with  respect  to  the  cone.  In  co  there  is  one 
line,  and  there  may  be  many  lines,  perpendicular  to  o. 
Through  o  and  a  line  o'  in  co  perpendicular  to  o  pass  a 
plane  tt,  and  let  I  and  I'  be  the  hnes  in  which  the  plane  tt 
cuts  the  cone.  Then  the  lines  o  and  o'  are  conjugates  with 
respect  to  I  and  I',  and,  being  perpendicular  to  each  other, 
they  bisect  the  angles  formed  by  I  and  I'.  Hence,  through 
any  line  o  there  is  one  plane  which  cuts  the  cone  in  lines 
that  form  an  angle  of  which  a  is  the  bisector. 

If  and  only  if  the  line  o  is  perpendicular  to  its  polar 
plane  co,  all  planes  through  o  cut  the  cone  in  lines  that 
make  an  angle  of  which  o  is  the  bisector.  In  this  case  the 
line  0  is  an  axis  of  symmetry  with  respect  to  the  cone. 

Manifestly  every  axis  of  symmetry  is  perpendicular  to 
its  polar  plane.  Also,  a  line  o'  parallel  to  o,  an  axis  of 
symmetry,  cuts  a  cone  in  two  points,  and  the  segment 
joining  these  points  is  bisected  by  the  polar  plane,  since 
the  axis  of  symmetry,  the  line  joining  the  vertex  to  the  in- 
tersection of  the  plane  co  with  o',  and  the  lines  I  and  V  are 
harmonic.  Hence  the  polar  plane  of  an  axis  of  symmetry 
is  a  plane  of  symmetry. 

It  can  be  shown  that  every  quadric  cone  has  one  axis 
Oj  of  symmetry  and  a  plane  co^  of  symmetry  which  is  per- 
pendicular to  it.  The  plane  Wj  cuts  the  cone  in  a  conic 
that  has  two  principal  axes  which  we  may  call  o^  and  Og. 
Of  the  lines  Oj,  o^,  o^  each  pair  is  conjugate  to  the  third 
line  and  determines  the  polar  plane  of  this  line.  Moreover, 
each  of  these  polar  planes  is  perpendicular  to  its  polar  line, 
and  hence  to  the  other  two  polar  planes.  There  are,  there- 
fore, three  axes  of  symmetry,  perpendicular  eaoli  to  each, 
and  three  planes  of  symmetry,  perpendicular  each  to  each. 


CYLINDERS 


171 


186.  Cylinders.  Hitherto  it  has  been  assumed  that  the 
axes  of  the  generating  axial  pencils  intersect  in  the  finite 
part  of  space.  If,  however,  the  axes  of  the  generating  axial 
pencils  are  parallel,  then  all  the  generating  lines  are  parallel 
to  them,  and  the  vertex  of  the  surface  is  at  infinity.  In 
this  case  the  surfaces  generated  are  called  cylinders. 


Hyperbolic  Cylinder      Parabolic  Cylinder      Elliptic  Cylinder 


Cylinders  are  classified  with  reference  to  their  relation 
to  the  plane  at  infinity.  The  plane  at  infinity  may  cut 
the  cylinder  in  two,  one,  or  no  straight  lines.  In  these 
cases  a  section  perpendicular  to  the  generating  lines  of 
the  cylinder  is  a  hyperbola,  a  parabola,  or  an  ellipse 
respectively;  and  the  cylinder  is  said  to  be  hyperholic^ 
parabolic^  or  elliptic,  as  the  case  may  be. 

In  the  case  of  a  cylinder  the  plane  at  infinity  and  cer- 
tain of  its  lines  are  in  the  bundle  to  which  the  cylinder 
belongs.  The  plane  at  infinity  has  a  polar  line  which  is 
an  axis  of  symmetry  and  is  called  the  axis  of  the  cylinder. 
It  can  be  shown  that  certain  planes  through  this  axis  and 
all  planes  perpendicular  to  it  are  planes  of  symmetry.  In 
the  case  of  the  parabohc  cylinder  the  polar  line  of  the 
plane  at  infinity  is  at  infinity  and  lies  in  the  cylinder. 


172  QUADRIC  CONES 

Exercise  34.    Quadric  Cones 

1.  Prove  the  theorem  regarding  quadric  cones  which  corre- 
sponds to  Steiner's  theorem  regarding  conies. 

2.  Every  conic  surface  of  the  second  class  is  of  the  second 
order.    Prove  also  the  converse. 

3.  The  hexahedral  angle  whose  faces  are  determined  by 
the  six  pairs  of  alternate  edges  of  another  hexahedral  angle 
which  is  inscribed  in  a  quadric  cone  has  its  faces  tangent  to  a 
quadric  cone. 

4.  Inscribe  in  a  quadric  cone  a  trihedral  angle  whose  three 
edges  shall  be  in  three  given  planes. 

5.  If  a  variable  simple  four-flat  so  moves  as  always  to  be 
circumscribed  about  a  given  quadric  cone,  while  three  of  its 
edges  move  each  in  one  of  three  fixed  coaxial  planes,  then 
the  fourth  edge  moves  on  a  fourth  fixed  plane  coaxial  with 
the  three  given  cones. 

6.  State  the  properties  of  polar  lines  and  planes  of  quadric 
cones  corresponding  to  those  of  poles  and  polars  of  conies  which 
are  given  in  §§  166  and  167. 

7.  Find  the  points  of  intersection  of  a  given  straight  line 
with  a  quadric  cone  of  which  five  determining  elements  are 
also  given. 

8.  In  a  bundle  a^,  fi^,  y^  and  a,^,  /S^,  y^  are  two  sets  of  fixed 
coaxial  planes.  Two  planes  tt^  and  tt^  so  move  that  the  lines 
determined  by  tt^  and  a^,  ir^  and  a^;  tt^  and  )8j,  tt^  and  /3.^', 
IT  and  yj,  tt^  and  y.^  lie  in  three  coaxial  planes.  Find  the  surface 
generated  by  the  line  common  to  tt^  and  tt.^. 

9.  In  a  bundle  the  edges  of  a  trihedral  angle,  whose  planes 
are  a,  /8,  y  and  which  is  self-polar  with  respect  to  a  given  quadric 
cone,  determine  with  any  line  o  the  planes  a',  ^',  y'.  If  the 
polar  plane  of  o  is  w,  the  pairs  of  lines  determined  by  w  with 
a  and  a',  by  w  with  ^  and  /3',  and  by  w  with  y  and  y'  form  a 
pencil  in  involution. 


CHAPTER  XIV 

SKEW  RULED  SURFACES 

187.  Skew  Ruled  Surfaces.  The  third  set  of  figures 
which  were  projectively  generated  was  found  to  consist  of 
the  ruled  surfaces  of  the  second  order  that  are  not  conic. 
Before  discussing  these  we  shall  consider  the  classification 
of  ruled  surfaces  in  general. 

One  classification  of  ruled  surfaces  is  based  upon  the 
law  governing  the  motion  of  the  generating  line.  At  any 
instant  the  motion  of  this  line  may  be  a  revolution  about 
one  of  its  o^vn  points  or  it  may  be  a  displacement  by 
virtue  of  which  the  line  immediately  ceases  to  intersect 
its  present  position.  In  the  former  case  it  is  sometimes 
said  that  every  pair  of  consecutive  generators  intersect, 
and  m  the  latter  case  it  is  said  that  no  two  consecutive 
generators  intersect.  Surfaces  generated  in  the  first  way 
are  called  developable  surfaces,  and  those  generated  in  the 
second  way  are  called  skeiv  surfaces.  Cones  and  cylinders 
are  examples  of  developable  surfaces,  but  they  are  of  a 
special  type,  inasmuch  as  each  of  their  generators  inter- 
sects every  other  one.  Likewise,  skew  ruled  surfaces  of  the 
second  order  and  second  class  are  special  in  character,  for 
no  generator  intersects  any  other  of  the  set,  even  though 
they  be  not  consecutive.  Generators  usually  cut  other 
generators  of  the  set  if  the  latter  are  not  consecutive. 

In  this  chapter  only  a  few  specially  interesting  facts 
regarding  the  surfaces  of  the  second  order  and  second 
class  will  be  established. 

173 


174 


SKEW  KULED  SUKEACES 


Theorem,  second  Set  of  Generating  Lines 

188.  Every  skew  ruled  surface  generated  hy  the  intersec- 
tions of  corresponding  planes  of  two  projective  axial  pencils 
has  also  a  second  set  of  generating  lines  whose  relation  to  the 
surface  is  similar  to  that  of  the  first  set.  Each  member  of 
either  set  of  generators  intersects  no  others  of  its  own  set,  but 
intersects  every  one  of  the  other  set.  Through  every  point  of 
the  surface  there  pass  two  generators,  one  of  each  set. 


Proof.  Every  plane  through  any  generatmg  line  a  of 
the  surface  cuts  the  surface  along  the  line  a  and  also 
along  a  second  line  a-^,  and  nowhere  else. 

Moreover,  the  line  a^  cuts  each  of  the  generating  lines 
that  have  been  noted.  The  infinitely  many  planes  through 
a  cut  the  surface  in  infinitely  many  lines  a^,  a^,  a^,  •  •  •, 
each  of  which  cuts  every  one  of  the  generators ;  and 
every  point  of  the  surface  lies  on  one  and  only  one  of 
the  new  lines. 

No  two  of  these  lines  intersect;  for  if  they  did,  all  the 
generators  would  lie  in  the  plane  determined  by  them. 


SECOND  SET  OF  GENERATING  LINES       175 

Consider  now  the  two  sets  of  points  B^^  B^,  ^3?  •  •  • 
and  Cj,  Cg,  Cg,  •  •  •  in  which  the  lines  a^,  ag,  a^,  -  •  •  intersect 
two  other  generators  b  and  c. 

These  sets  of  points  are  the  intersections  of  the  gener- 
ators b  and  c  with  the  planes  of  the  axial  pencil  whose 
base  is  a  and  whose  planes  pass  through  the  lines  a^,  a^, 
flg,  • .  • ;  and  consequently  they  constitute  perspective  (but 
not  coplanar)  ranges. 

Then  the  axial  pencil  whose  base  is  b  and  whose  planes 
pass  through  Cj,  Cj,  Cg,  •  •  •  and  the  axial  pencil  whose 
base  is  c  and  whose  planes  pass  through  B^,  B^j  -Sg,  •  •  •, 
being  perspective  respectively  with  the  ranges  C^C^C^' '  • 
and  B^B^B^  •  •  •,  are  projective  with  each  other. 

Corresponding  planes  of  these  axial  pencils  intersect  in 
the  lines  a^,  a^.,  ag,  •  •  •,  which  are  therefore  generating 
lines  of  a  skew  quadric  ruled  surface. 

The  latter  skew  ruled  surface  must  coincide  with  the 
original  one,  since  the  lines  a^,  a^,  ^g,  •  •  •  contain  all  the 
points  of  the  original  ruled  surface,  and  no  others.  Hence 
these  lines  must  constitute  a  second  set  of  generators  for 
that  ruled  surface. 

189.  Corollary.  The  lines  of  either  set  of  generators 
determine  projective  ranges  on  any  ttco  lines  of  the  other  set. 

190.  Conjugate  Reguli.  Two  reguli  which  are  related  as 
are  the  two  in  §  188  are  called  conjugate  reguli. 

The  theorem  of  §  188  may  be  restated  as  a  corollary  to 
this  definition  as  shown  below. 

191 .  Corollary.  Every  skew  ruled  surface  of  the  second 
order  carries  two  conjugate  reguli.  Each  line  of  either  regulus 
intersects  no  lines  of  its  own  regulus,  but  intersects  each  line 
of  the  other  regulus.  Through  every  point  of  the  surface  there 
pass  two  lines,  one  from  each  regulus. 


176  SKEW  KULED  SURFACES 

Theorem.  Determination  by  Generators 

192.  Given  three  straight  lines  no  two  of  which  are  coplatiar, 
there  exists  one  and  only  one  skew  quadric  ruled  surface  of  which 
each  of  these  lines  is  a  generator. 

A, 


Proof.  Let  a,  b,  c  be  three  straight  hues  no  two  of  which 
are  coplanar. 

Through  each  point  of  a  one  line  and  only  one  can  be 
.drawn  to  meet  all  three  of  the  lines. 

Let  p^,  j»2,  and  p^  be  three  lines  which  meet  a,  b,  c. 

Then  p^  together  with  the  three  lines  a,  b,  c,  and  p^ 
together  with  the  same  lines,  determine  triads  of  planes  of 
axial  pencils  whose  bases  are  p^  and  p^.  These  triads 
determine  a  projectivity  between  the  pencils,  and  this 
projectivity  determines  one  skew  quadric  ruled  surface  of 
which  a,  b,  c  are  generators. 

Any  two  corresponding  planes  of  the  projective  axial 
pencils  intersect  in  a  line  that  meets  p^  and  p^.  This  line 
also  meets  p^;  for  if  p^  meets  a,  b,  c  in  Jg,  B^,  Cg  respec- 
tively, the  two  axial  pencils  above  mentioned  are  each  per- 
spective with  the  same  range  on  jOg,  the  perspectivities  being 
determined  by  the  correspondence  of  A^,  B^,  Cg  to  the  planes 
p^a,  p^b,  p^e  in  the  one  case,  and  the  planes  p,^a^  p^b,  p.^c  in 
the  other.  It  follows  that  corresponding  planes  cut  p^  in 
the  same  point,  and  hence  their  line  of  intersection  cuts  py 

Accordingly,  the  surface  is  uniquely  determined  by  the 
three  generators  a,  b,  c. 


CLASSIFICATION  177 

imiimniiipi^^ 


Hyperbolic  Paraboloid,  the  conjugate  reguli  being  formed 
by  rods.  Certain  sections  are  parabolas,  other  sections 
are  hyperbolas.  The  curvature  of  the  surface  is  not 
secured  by  the  pressure  of  one  set  of  rods  upon  the  other 


Hyperboloid  of  One  Sheet,  the  conjugate  reguli  being 
formed  by  straight  rods.  The  surface  in  the  neighbor- 
hood of  its  center  of  symmetry  is  shown.  Horizontal 
sections  are  ellipses,  vertical  sections  are  hyperbolas 


gjjjjjpjjjM^^ 


178  SKEW  RULED  SURFACES 

193.  Skew  Quadric  Ruled  Surfaces  Classified.  These  sur- 
faces are  classified  according  to  the  nature  of  their  sections 
by  the  plane  at  infinity.  As  has  been  shown,  any  plane 
section  of  one  of  these  surfaces  is  a  conic  and  degenerates 
into  two  straight  lines  if  the  cutting  plane  contains  a 
generator.    The  surfaces  are,  therefore,  of  two  sorts: 

1.  Hyperbolic  paraboloids,  or  those  whose  intersections 
with  the  plane  at  infinity  are  pairs  of  generators. 

2.  Hyperboloids  of  one  sheet,  or  those  whose  intersections 
with  the  plane  at  infinity  are  nondegenerate  conies. 

It  may  be  noted  that  if  a  hyperbolic  paraboloid  is  regarded 
as  generated  by  the  joining  lines  of  corresponding  points  of 
two  projective  ranges,  the  points  at  infinity  of  the  ranges 
are  found  to  be  corresponding  points.  Hence  in  this  case 
(and  in  this  case  only)  the  ranges  are  similar.  Accord- 
ingly, if  corresponding  points  of  two  similar  (but  not 
coplanar)  ranges  are  connected  by  threads,  a  good  model 
of  a  hyperbolic  paraboloid  may  be  constructed. 

To  exhibit  both  sets  of  generating  lines  it  is  better  to  use  a 
quadrilateral  A  BCD  hinged  at  two  opposite  vertices,  as  B  and  D,  so 
that  the  triangles  ABD, 
CBD  can  be  adjusted  to 
lie  in  different  planes. 
Congruent  ranges  can  be 
taken  on  A  B  and  CD  and 
also  on  BC  and  DA.  Cor- 
responding points  can  be 
joined  by  strings,  and  in 
this  manner  an  excellent 
model  can  be  constructed 
with  very  little  trouble. 

Directions  for  constructing  a  string  model  of  the  hyperboloid  of 
one  sheet  are  not  so  easily  given.  The  existence  of  such  a  surface  is 
evident,  since  it  is  generated  by  the  lines  joining  corresponding 
points  of  projective  ranges  which  are  not  coplanar  and  not  similar. 


CLASSIFICATION 

OmmiiJliiiii^^ 


179 


Hyperbolic  Paraboloid,  the  conjugate  reguli  being  formed 

by  straight  rods.     The    surface  near  the  vertex,   or 

saddle  point,  is  shown.    Certain  sections  through  the 

vertex  are  parabolas,  others  are  hyperbolas 


Hyperbolic  Paraboloid,  the  conjugate  reguli  being  formed 

by  strings,  ff  every  string  in  either  set  is  cut,  the  strings 

in  the  other  set  retain  their  positions 


180  SKEW  RULED  SURFACES 

Exercise  35.   Skew  Ruled  Surfaces 

1.  A  regulus  is  determined  by  two  nonintersecting  lines 
and  three  noncollinear  points,  no  two  of  which  are  coplanar 
with  any  of  the  lines. 

2.  If  a  regulus  contains  a  line  at  infinity,  the  conjugate 
regulus  also  contains  a  line  at  infinity. 

3.  Determine  three  pairs  of  quadric  ruled  surfaces  which 
have  in  common  two  given  noncoplanar  lines  and  also  respec- 
tively no,  one,  and  two  generators  of  the  other  set. 

4.  Determine  a  quadric  ruled  surface  which  contains  two 
given  noncoplanar  lines  and  a  given  point  exterior  to  them. 
How  many  such  surfaces  are  there  ?  Find  any  additional  lines 
which  are  common  to  such  surfaces. 

5.  If  four  generators  of  a  regulus  cut  one  generator  of 
the  conjugate  regulus  in  a  harmonic  range,  they  cut  every 
generator  of  the  conjugate  regulus  in  a  harmonic  range. 

Four  generators  of  a  regulus  which  have  the  property  mentioned  in 
Ex.  6  are  called  harmonic  generators. 

6.  Given  any  three  lines  in  space,  no  two  of  which  are 
coplanar,  find  a  fourth  line  which,  with  the  three  given  lines, 
constitutes  a  set  of  harmonic  generators  of  a  regulus. 

7.  If  a  line  so  moves  as  constantly  to  intersect  each  of  two 
noncoplanar  lines  and  also  to  remain  parallel  to  a  given  plane, 
the  line  generates  a  hyperbolic  paraboloid. 

8.  If  a  range  and  a  flat  pencil  which  do  not  lie  in  the  same 
plane  or  in  parallel  planes  are  projective,  and  if  from  each 
point  of  the  range  a  line  is  drawn  parallel  to  the  correspond- 
ing line  of  the  flat  pencil,  these  parallel  lines  all  lie  on  a 
hyperbolic  paraboloid. 

9.  The  locus  of  the  harmonic  conjugates  of  any  point  with 
respect  to  a  ruled  surface  is  a  plane. 

10.  The  lines  (or  planes)  of  any  bundle  which  are  tangent 
to  a  quadric  ruled  surface  generate  a  quadric  cone. 


HISTORY  OF  PROJECTIVE  GEOMETRY 

The  history  of  geometry  may  be  divided  roughly  into 
four  periods:  (1)  The  synthetic  geometry  of  the  Greeks, 
including  not  merely  the  geometry  of  Euclid  but  the 
work  on  conies  by  Apollonius  and  the  less  formal  contri- 
butions of  many  other  writers;  (2)  the  birth  of  analytic 
geometry,  in  which  the  synthetic  geometry  of  Desargues, 
Kepler,  Roberval,  and  other  writers  of  the  seventeenth 
century  merged  into  the  coordinate  geometry  of  Descartes 
and  Fermat ;  (3)  the  application  of  the  calculus  to  geom- 
etry,—  a  period  extending  from  about  1650  to  1800,  and 
including  the  names  of  Cavalieri,  Newton,  Leibniz,  the  Ber- 
noullis,  L'HSpital,  Clairaut,  Euler,  Lagrange,  and  D'Alem- 
bert,  each  one,  especially  after  Cavalieri,  being  primarily 
an  analyst  rather  than  a  geometer ;  (4)  the  renaissance 
of  pure  geometry,  beginning  with  the  nineteenth  century 
and  characterized  by  the  descriptive  geometry  of  Monge, 
the  projective  geometry  of  Poncelet,  the  modern  synthetic 
geometry  of  Steiner  and  Von  Staudt,  the  modern  analytic 
geometry  of  Pliicker,  the  non-Euclidean  hypotheses  of 
Lobachevsky,  Bolyai,  and  Riemann,  and  the  laying  of  the 
logical  foundations  of  geometry,  —  a  period  of  remarkable 
richness  in  the  development  of  all  phases  of  the  science. 

It  is  in  this  fourth  period  that  projective  geometry  has 
had  its  development,  even  if  its  origin  is  more  remote. 
The  origin  of  any  branch  of  science  can  always  be  traced 
far  back  in  human  history,  and  this  fact  is  patent  in  the 
case  of  tliis  phase  of  geometry.    The  idea  of  the  projection 

181 


182  •  HISTORY 

of  a  line  upon  a  plane  is  very  old.  It  is  involved  in  the 
treatment  of  the  intersection  of  certam  surfaces,  due  to 
Archytas,  in  the  fifth  century  B.C.,  and  appears  in  various 
later  works  by  Greek  writers.  Similarly,  the  invariant  prop- 
erty of  the  anharmonic  ratio  was  essentially  recognized 
both  by  Menelaus  in  the  first  century  a.d.  and  by  Pappus 
in  the  third  century.  The  notion  of  infinity  was  also  famil- 
iar to  several  Greek  geometers,  so  that  various  concepts 
that  enter  into  the  study  of  projective  geometry  were  com- 
mon property  long  before  the  science  was  really  founded. 

One  of  the  first  important  steps  to  be  taken  in  modern 
times,  in  the  development  of  this  form  of  geometry,  was 
due  to  Desargues,  a  French  architect.  In  a  work  on  conic 
sections,  published  in  1639,  Desargues  set  forth  the  founda- 
tion of  the  theory  of  four  harmonic  points,  not  as  done 
today,  but  based  on  the  fact  that  the  product  of  the  dis- 
tances of  two  conjugate  pohits  from  the  center  is  con- 
stant. He  also  treated  of  the  theory  of  poles  and  polars, 
although  not  using  these  terms.  In  1640  Pascal,  then  only 
a  youth  of  sixteen,  published  a  brief  essay  on  conies  setting 
forth  the  well-known  theorem  that  bears  his  name. 

The  descriptive  geometry  of  Monge  is  a  kind  of  pro- 
jective geometry,  although  it  is  not  what  we  ordinarily 
mean  by  this  term.  He  was  a  French  geometer  of  the 
period  of  the  Revolution,  and  had  been  in  possession  of 
his  theory  for  over  thirty  years  before  the  publication  of 
his  "Geometrie  Descriptive  "  (1795).  It  is  true  that  certain 
of  the  features  of  this  work  can  be  traced  back  to  De- 
sargues, Taylor,  Lambert,  and  Frezier,  but  it  was  Monge 
who  worked  out  the  theory  as  a  science.  Inspired  by  the 
general  activity  of  the  period,  but  following  rather  in  the 
steps  of  Desargues  and  Pascal,  Carnot  treated  chiefly  of 
the  metric  relations  of  figures.   In  particular  he  investigated 


HISTORY  183 

these  relations  as  connected  with  the  theory  of  transver- 
sals, —  a  theory  whose  fundamental  property  of  a  four- 
rayed  pencil  goes  back  to  Pappus,  and  which,  though 
revived  by  Desargues,  was  set  forth  for  the  first  time  in 
its  general  form  by  Carnot  in  his  "  Geometric  de  Posi- 
tion "  (1803),  and  supplemented  in  his  "  Theorie  des 
Transversales  "  (1806).  In  these  works  Carnot  introduced 
negative  magnitudes,  the  general  quadrilateral,  the  general 
quadrangle,  and  numerous  other  similar  features  of  value 
to  the  elementary  geometry  of  today. 

Projective  geometry  had  its  origin  somewhat  later  than 
the  period  of  Monge  and  Carnot.  Newton  had  discovered 
that  all  curves  of  the  third  order  can  be  derived  by  central 
projection  from  five  fundamental  types.  But  in  spite  of  this 
the  theory  attracted  so  little  attention  for  over  a  century 
that  its  origin  is  generally  ascribed  to  Poncelet.  A  pris- 
oner in  the  Russian  campaign,  confined  at  Saratoff  on  the 
Volga  (1812-1814),  "  prive,"  as  he  says,  "  de  toute  espece 
de  livres  et  de  secours,  surtout  distrait  par  les  malheurs 
de  ma  patrie  et  les  miens  propres,"  Poncelet  still  had  the 
vigor  of  spirit  and  the  leisure  to  conceive  the  great  work, 
"  Traite  des  Propri6t^s  Projectives  des  Figures,"  which  he 
published  in  1822.  In  this  work  was  first  made  promi- 
nent the  power  of  central  projection  in  demonstration  and 
the  power  of  the  principle  of  continuity  in  research.  His 
leading  idea  was  the  study  of  projective  properties,  and 
as  a  foundation  principle  he  introduced  the  anharmonic 
ratio, —  a  concept,  however,  which  dates  back  to  Menelaus 
and  Pappus,  and  which  Desargues  had  also  used.  Mobius, 
following  Poncelet,  made  much  use  of  the  anharmonic 
ratio  in  his  "  Barycentrische  Calcul"  (1827),  but  he  gave 
it  the  name  Doppelschnitt-l'erhaltniss  (^ratio  hisectionalis), 
a  term  now  in  common  use  under  Steiner's    abbreviated 


184  HISTORY 

form  Doppelverhdltniss.  The  name  anharmonie  ratio  or 
anharmonie  function  (rapport  anharmonique,  or  fonetion 
anharmonique)  is  due  to  Chasles,  and  cross-ratio  was  sug- 
gested by  Clifford.  The  anharmonie  point-and-line  prop- 
erties of  conies  have  been  elaborated  by  Brianchon,  Chasles, 
Steiner,  Dupin,  Hachette,  Gergonne,  and  Von  Staudt.  To 
Poncelet  is  due  the  theory  of  figures  homologiques,  the  per- 
spective axis  and  perspective  center  (called  by  Chasles 
the  axis  and  center  of  homology),  an  extension  of  Carnot's 
theory  of  transversals,  and  the  cordes  ideales  of  conies,  which 
Pliicker  applied  to  curves  of  all  orders.  Poncelet  also 
discovered  what  Salmon  has  called  "  the  circular  points  at 
infinity,"  thus  completing  and  establishing  the  first  great 
principle  of  modem  geometry,  — the  principle  of  continuity. 
Brianchon  (1806),  tlirough  his  application  of  Desargues's 
theory  of  polars,  completed  the  foundation  ijvhich  Monge 
had  begun  for  Poncelet's  theory  of  reciprocal  polars  (1829). 

Steiner  (1832)  gave  the  first  complete  discussion  of  the 
projective  relations  between  rows,  pencils,  etc.,  and  laid  the 
foundation  for  the  subsequent  development  of  pure  geom- 
etry. He  practically  closed  the  theory  of  conic  sections,  of 
the  corresponding  figures  in  three-dimensional  space,  and 
of  surfaces  of  the  second  order.  With  him  opens  the  period 
of  special  study  of  curves  and  surfaces  of  higher  order.  His 
treatment  of  duality  and  his  application  of  the  theory  of  pro- 
jective pencils  to  the  generation  of  conies  are  masterpieces. 

Cremona  began  his  publications  in  1862.  His  elementary 
work  on  projective  geometry  (1875)  is  familiar  to  English 
readers  in  Leudesdorf  s  translation.  The  recent  contribu- 
tions have  naturally  been  of  an  advanced  character,  seek- 
ing to  lay  more  strictly  logical  foundations  for  the  science, 
and  in  this  line  the  American  work  by  Professors  Veblen 
and  Young  is  noteworthy. 


INDEX 


PAGE 

Anharmonic  ratio  .    .     21,  24, 183 

Asymptote 144 

Axial  pencil 8,  96 

Axial  projection 3 

Axis,  of  a  cylinder     ....    171 

of  homology 37 

of  projection 3 

of  symmetry 170 

Base 8 

Brianchon  point 122 

Brianchon  theorem  .  .  121, 148 
Bundle 8 

Center,  of  a  conic 163 

of  homology 37 

of  involution 74 

of  projection 2 

Central  projection  .....        2 

Class  of  a  figure 80 

Classification,  of  conies .    .    .    143 

of  prime  forms    ....      10 

of  projectivities  ....      60 

Congruent  elements   ....      72 

Congruent  pencils 59 

Congruent  ranges  .    .    .    .    59, 68 

Conic 109, 115, 143,  156 

Conjugate  reguli 175 

Conjugates  ...  31,  72,  157, 164 
Constant  of  homology  ...  39 
Constructions  of  second  order  135 
Correlative  propositions     .    .      15 

Cylinder 171 

185 


PAOB 

Desargues's  theorem  ....  131 
Descriptive  properties  ...  21 
Developable  surface  ....    173 

Diagonal  lines 34 

Diagonal  points 33 

Diagonal  triangle   .    .    .    .    33, 34 

Diameter 163 

Directrix 93 

Double  elements 64 

Duality 15,  81,  162 

Elements  at  infinity   ...    5, 144 

Ellipse 143 

Elliptic  cylinder 171 

Elliptic  projectivity  ....  66 
Envelope 80 

Figures  of  second  order  .  .  101 
Flat  pencil  .    .    8,  59,  89,  90,  96,  97 

Four-point 33 

Four-side 34 

Fundamental  theorem    ...      44 

Generation  of  a  figure  ...  80 
Generator 93 

Harmonic  conjugates     .    .31, 151 

Harmonic  form 31,  36 

Harmonic  homology  ....      39 

Harmonic  pencil 31 

Harmonic  range 31 

Hexagon 120 

Homology 37 


186 


INDEX 


PAGE 

Hyperbola 143 

Hyperbolic  cylinder  ....  171 

Hyperbolic  involution    ...  73 

Hyperbolic  paraboloid   ...  178 

Hyperbolic  projectivity ...  66 

Hyperboloids 178 

Infinity 5, 144 

Involution 72,  74 

Line  at  infinity 5 

Line  involution 76 

Locus 79 

Metric  properties 21 

One-dimensional  forms  ...  8 

One-to-one  correspondence    .  10 

Opposite  sides 33,  120 

Opposite  vertices 34 

Order,  of  a  construction     .    .  135 

of  a  figure 80 

Orthogonal  projection    ...  1 

Parabola 143 

Parabolic  cylinder 171 

Parabolic  projectivity     ...  66 

Parallel  projection 2 

Pascal  line 122 

Pascal's  theorem    .    .    .    121,  147 

Pencil 8 

Perspectivity 11 

Plane,  at  infinity 5 

of  points 8 

of  symmetry 1 70 

Plane  figures 101 

Point  at  infinity 5 

Point  involution 75 

Polar 151,  154 

Polar  reciprocal 162 


PAQB 

Pole 151,  166 

Prime  forms 8 

Principal  axes 164 

Principal  diameter  ....  164 
Principle  of  duality    .    .      15,  162 

Projection i,  7 

Projectivity 41,  66 

Projector 3,  7 

Quadrangle 33 

Quadric 110 

Quadric  cone 167 

Quadric  surface  ...  93,  110, 178 
Quadrilateral 34 

Kange  of  points 8,  96 

Reciprocity 16 

Regulus 93,  176 

Relation,  of  angles     ....      23 
of  anharmonic  ratios  .    .      26 

of  segments 22 

Ruled  surface 93 

Section 3 

Self-conjugate  triangle  .  .  .  158 
Self-corresponding  elements  .  64 
Self -polar  triangle  .    .    .    .    .    158 

Sense 22,23 

Sheaf 8 

Similar  figures 39 

Similar  ranges 69 

Similitude,  ratio  of  ...  .  39 
Skew  surface  .    .  93,  111,  173,  178 

Steiner's  theorem 104 

Superposed  forms 63 

Symbols 1,  H,  41 

Ten  prime  forms 8 

Three-dimensional  forms  .  .  8 
Two-dimensional  forms ...        8 


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